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### Rotating Rings Bodies Stress Equations and Calculator

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Rotating Rings Bodies Stress Equations and Calculator

Hollow Disk Ring Rotating at *ω* rad/s

Preview: Rotating Ring Bodies Stress Calculator

- The outside radius of the ring, or disk, is large compared with the thickness: r
_{o}≥ 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

*σ _{t} = ρ ω^{2} [ ( 3 + v ) / 8 ] [ r_{i}^{2} + r_{o}^{2} + r_{i}^{2} r_{o}^{2} / r^{2} - ( 1 + 3 v ) / ( 3 + v ) r^{2} ] *

The general expression for the radial stress of a rotating disk of uniform thickness

Eq. 2

*σ _{r} = ρ ω^{2} [ ( 3 + v ) / 8 ] [ r_{i}^{2} + r_{o}^{2} - r_{i}^{2} r_{o}^{2} / r^{2} - r^{2} ] *

The maximum radial stress occurs at r^{2} = r_{i} r_{o}

Eq. 3

*σ _{r} = ρ ω^{2} ( r_{o} - r_{i} )^{2} ( 3 + v ) / 8 *

The maximum tangential stress occurs at inner boundary where r = r_{i}

*σ _{θ(max)} = [ ( 3 + v ) / 4 ] ρ ω^{2} ( r_{o}^{2} + ( 1 - v ) / ( 3 + v ) ) r_{i}^{2}*

For a rotating disk, use *r _{i}* = 0 in these equations.

where

r = radius to the stress element under consideration, m (in)

ρ = mass density, kg/m^{3} (lbm/in^{3})

ω = angular velocity of the ring, rad/s

*v* = Poisson's ratio

*r _{i}* = inside radius, m (in)

*r*= outside radius, m (in)

_{o}For a rotating disk, use r_{i} = 0 in these equations.

Figure 1 Rotating disk of uniform thickness.

Figure 2 Simplified Rotating Ring Stress

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