Rotating Solid Cylinder Stress Equations and Calculator

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Solid Cylinder Rotating at ω rad/s

The stresses are

The tangential stress equation
Eq. 1
σ_θ = ρ ω² / ( 8 ( 1 - v ) ) [ ( 3 - 2v ) r_o² - ( 1 + 2v ) r² ]

The radial stress equation
Eq. 2
σ_r = ρ ω² / 8 [ ( 3 - 2 v ) / ( 1 - v ) ] ( r_o² - r_i² )

The maximum stress occurs at the center
Eq. 3
σ_r(max) = σ_θ(max) = ρ ω² / 8 ( ( 3 - 2 v ) / ( 1 - v ) ) r_o²

The axial strain in the z direction (ends free) equation
Eq. 4
ε_z = -v / 2 ( ρ ω² r_o² ) / E

The axial stress under plane strain condition (ends free) equation
Eq. 5
σ_z = ρ ω² / 4 ( v / ( 1 - v ) ( r_o² - 2r² )

The axial stress under plane strain condition (ends constrained)
Eq. 6
σ_z = ρ ω² v / ( 4( 1 - v ) [ 0.5 ( 3 - 2v) r_o² - 2r² ]

Figure 1 Rotating Solid Cylinder Stress

where

E = modulus of elasticity or Young’s modulus, Pa [psi (lb/in²)]
r = radius to the stress element under consideration, m (in)
r_o = outside radius, m (in)
ρ = mass density, kg/m³ (lbm/in³)
ω = angular velocity of the ring, rad/s
v = Poisson's ratio

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