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### Hollow Cylinder Rotating Stress Equations and Calculator

Hollow Cylinder Rotating at ω rad/s Stress Equations and Calculator

The stresses are

The tangential stress at any radius r formula
Eq. 1
σθ = ρ ω2 / 8 * ( ( 3 - 2v ) / ( 1 - v ) ) [ ri2 + ro2 + ri2 ro2 / r2 - ( ( 1 + 2v ) / ( 3 - 2v ) ) r2 ]

Eq. 2
σr = ρ ω2 / 8 ( 3 - 2 v ) / ( 1 - v ) [ ri2 + ro2 - ri2 ro2 / r2 - r2 ]

The axial stress (ends free) at any radius r formula
Eq. 3
σz = ρ ω2 / 4 ( v / ( 1 - v ) (ri2 + ro2 - 2r2 )

The axial strain in the z direction (ends free) equation
Eq. 4
σz = v ρ ω2 / 4 ( 3 - 2v ) / ( 1 - v ) [ ri2 + ro2 - 2 r2 / ( 3 - 2 v) ]

The maximum stress occurs at the inner surface where r = ri formula
Eq. 5
σθ(max) = ρ ω2 / 4 ( 3 - 2v ) / ( 1 - v ) [ ro2 + ( ( 1 - 2v ) / ( 3 - 2 v) ri2 ] )

The axial strain in the z direction (ends free) formula
Eq. 6
εz = - v ρ ω2 / (2 E ) ( ri2 + ro2 )

The displacement u at any radius r of a thin hollow rotating disk formula
Eq. 7
u = [ ρ ω2 r / E ( ( 3 + v ) ( 1 - v )) / 8 ( ro2 + ri2 + ( 1 + v ) / ( 1 - v) ro2 ri2 / r2 - ( 1 + v ) / ( 3 + v ) r2 ) ]

Figure 1

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:

Machine Design Handbook 2nd Edition, 2001
Prof. Dr. K. Lingaiah
Former Principal, Professor and Head, Department of Mechanical Engineering
University Visvesvaraya College of Engineering, Bangalore University, Bangalore