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### Rotating Solid Cylinder Stress Equations and Calculator

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Strength and Mechanics of Materials Menu

Rotating Solid Cylinder Stress Equations and Calculator

Solid Cylinder Rotating at *ω* rad/s

Preview: Rotating Solid Cylinder Stress Calculator

The stresses are

The tangential stress equation

Eq. 1

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

The radial stress equation

Eq. 2

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

The maximum stress occurs at the center

Eq. 3

*σ _{r(max)} = σ_{θ(max)} = ρ ω^{2} / 8 ( ( 3 - 2 v ) / ( 1 - v ) ) r_{o}^{2}*

The axial strain in the z direction (ends free) equation

Eq. 4

*ε _{z} =
-v / 2 ( ρ ω^{2} r_{o}^{2} ) / E*

The axial stress under plane strain condition (ends
free) equation

Eq. 5

*σ _{z} =*

*ρ ω*

^{2}/ 4 ( v / ( 1 - v )*( r*

_{o}^{2}- 2r^{2})The axial stress under plane strain condition (ends
constrained)

Eq. 6

*σ _{z} = ρ ω^{2} v / ( 4( 1 - v ) [ 0.5 ( 3 - 2v) r_{o}^{2} - 2r^{2} ] *

Figure 1 Rotating Solid Cylinder Stress

where

*E* = modulus of elasticity or Young’s modulus, Pa
[psi (lb/in^{2})]

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

*r _{o}* = outside radius, m (in)

*ρ*= mass density, kg/m

^{3}(lbm/in

^{3})

*ω*= angular velocity of the ring, rad/s

*v*= Poisson's ratio

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