Rotating Disk with a Central Hole

Stresses in a thin uniform disk with a central hole spinning at constant speed (Lamé-type closed form): radial and hoop stress at any radius, the peak hoop stress at the hole edge, the peak radial stress, the radial growth, and the stress concentration of the hole relative to a solid disk (which tends to 2 for a small hole).

  • Closed-form radial and hoop stresses in a spinning annular disk

  • Peak hoop stress at the bore, peak radial stress at r = √(ab), bore and rim growth

  • Stress concentration of the central hole relative to the solid disk

  • Stress distribution plot across the radius

Disk Geometry, Material and Speed

Unit System: /

Convention

Rotating disk with a central hole: inner radius a, outer radius b, speed omega, point P at radius r

Geometry




Material and Speed





Rotating Disk With Hole
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Rotating Disk with a Central Hole

Rotating Disk with a Central Hole

Problem Statement

A thin uniform disk of density ρ, inner radius a and outer radius b spins at a constant angular speed ω. The centrifugal body force ρω²r is carried by radial and hoop stresses that depend only on r. Wanted are the stresses at a radius, the peak values and the growth of the bore and rim.

Rotating disk with a central hole

Key Assumptions

Thin disk (plane stress), uniform thickness and density, linear elastic, free bore and rim, constant speed. A hub press-fit or blade load on the rim adds a Lamé solution that superposes; a thick disk, a tapered profile or a thermal gradient are different problems.

Solution

With q = (3 + ν)ρω²/8 the equilibrium equation d(rσr)/dr − σθ + ρω²r² = 0 with compatibility gives

σr = q(a² + b² − a²b²/r² − r²),    σθ = q[a² + b² + a²b²/r² − (1 + 3ν)r²/(3 + ν)],

which vanish in σr at r = a and r = b. The radial stress peaks at r = √(ab) with the value q(b − a)²; the hoop stress peaks at the bore, σθ(a) = (3 + ν)ρω²[b² + (1 − ν)a²/(3 + ν)]/4, where σr = 0, so the bore is in uni-axial hoop tension and the von Mises stress there equals the hoop stress. The radial growth is u = r(σθ − νσr)/E.

Stress Concentration of the Hole

A solid disk has its peak at the center, σ = qb², with equal radial and hoop stress. The hole doubles the hoop stress of the solid disk even when it is vanishingly small, and raises it further with size: K = σθ(a)/(qb²) = 2 + 2(1 − ν)(a/b)²/(3 + ν). This is the rotating-disk counterpart of the equal-biaxial Kirsch factor 2: at the center of a solid disk the stress state is equal biaxial, and a small hole there behaves as in a plate under equal biaxial tension.

Limits

  • The disk must be thin compared with its radius; the plane-stress solution ignores the small axial stress of thick disks.
  • Bursting speed: the bore yields first, at σθ(a) = Fty; the average hoop stress at burst is a separate (limit-load) calculation.


References

Timoshenko, S. P., and Goodier, J. N. (1970). Theory of Elasticity, 3rd ed. McGraw-Hill, New York. Chapter 4, article "Rotating disks": the plane-stress solution for the solid and the annular disk, the peak hoop stress at the bore and its limit of twice the solid-disk stress for a small hole.

Boresi, A. P., and Schmidt, R. J. (2003). Advanced Mechanics of Materials, 6th ed. John Wiley & Sons. Thick-wall cylinders and rotating disks: the same distributions with the radial growth.

Pilkey, W. D., and Pilkey, D. F. (2008). Peterson's Stress Concentration Factors, 3rd ed. John Wiley & Sons, Hoboken. Chapter 4, Section 4.3.8, "Rotating disk with a hole": the factor 2 + 2(1 − ν)(a/b)²/(3 + ν) on the solid-disk stress.

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