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### Rotating Solid Cylinder Stress Equations and Calculator

Rotating Solid Cylinder Stress Equations and Calculator

Solid Cylinder Rotating at ω rad/s

The stresses are

The tangential stress equation
Eq. 1
σθ = ρ ω2 / ( 8 ( 1 - v ) ) [ ( 3 - 2v ) ro2 - ( 1 + 2v ) r2 ]

Eq. 2
σr = ρ ω2 / 8 [ ( 3 - 2 v ) / ( 1 - v ) ] ( ro2 - ri2 )

The maximum stress occurs at the center
Eq. 3
σr(max) = σθ(max) = ρ ω2 / 8 ( ( 3 - 2 v ) / ( 1 - v ) ) ro2

The axial strain in the z direction (ends free) equation
Eq. 4
εz = -v / 2 ( ρ ω2 ro2 ) / E

The axial stress under plane strain condition (ends free) equation
Eq. 5
σz = ρ ω2 / 4 ( v / ( 1 - v ) ( ro2 - 2r2 )

The axial stress under plane strain condition (ends constrained)
Eq. 6
σz = ρ ω2 v / ( 4( 1 - v ) [ 0.5 ( 3 - 2v) ro2 - 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)
ro = outside radius, m (in)
ρ = mass density, kg/m3 (lbm/in3)
ω = angular velocity of the ring, rad/s
v = Poisson's ratio

Related:

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