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### Rotating Rings Bodies Stress Equations and Calculator

Rotating Rings Bodies Stress Equations and Calculator

Hollow Disk Ring Rotating at ω rad/s

• The outside radius of the ring, or disk, is large compared with the thickness: ro ≥ 10t.
• The thickness of the ring or disk is constant.
• The stresses are constant over the thickness.

The stresses are

The general expression for the tangential stress of a rotating disk of uniform thickness
Eq. 1
σt = ρ ω2 [ ( 3 + v ) / 8 ] [ ri2 + ro2 + ri2 ro2 / r2 - ( 1 + 3 v ) / ( 3 + v ) r2 ]

The general expression for the radial stress of a rotating disk of uniform thickness
Eq. 2
σr = ρ ω2 [ ( 3 + v ) / 8 ] [ ri2 + ro2 - ri2 ro2 / r2 - r2 ]

The maximum radial stress occurs at r2 = ri ro
Eq. 3
σr = ρ ω2 ( ro - ri )2 ( 3 + v ) / 8

The maximum tangential stress occurs at inner boundary where r = ri

σθ(max) = [ ( 3 + v ) / 4 ] ρ ω2 ( ro2 + ( 1 - v ) / ( 3 + v ) ) ri2

For a rotating disk, use ri = 0 in these equations.

where

r = radius to the stress element under consideration, m (in)
ρ = mass density, kg/m3 (lbm/in3)
ω = angular velocity of the ring, rad/s
v = Poisson's ratio
ri = inside radius, m (in)
ro = outside radius, m (in)

For a rotating disk, use ri = 0 in these equations.

Figure 1 Rotating disk of uniform thickness.

Figure 2 Simplified Rotating Ring Stress

Related:

Source: Shigley, Joseph E., and Charles R. Mischke, Mechanical Engineering Design, 5th ed., New York: McGraw-Hill, 1989.