Hollow Cylinder Rotating Stress Equations and Calculator

Related Resources: calculators

Hollow Cylinder Rotating Stress Equations and Calculator

Engineering Analysis
Strength and Mechanics of Materials Menu

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

Preview: Hollow Cylinder Rotating Stress Calculator

The stresses are

The tangential stress at any radius r formula
Eq. 1
σ_θ = ρ ω² / 8 * ( ( 3 - 2v ) / ( 1 - v ) ) [ r_i² + r_o² + r_i² r_o² / r² - ( ( 1 + 2v ) / ( 3 - 2v ) ) r² ]

The radial stress at any radius r formula
Eq. 2
σ_r = ρ ω² / 8 ( 3 - 2 v ) / ( 1 - v ) [ r_i² + r_o² - r_i² r_o² / r² - r² ]

The axial stress (ends free) at any radius r formula
Eq. 3
σ_z = ρ ω² / 4 ( v / ( 1 - v ) (r_i² + r_o² - 2r² )

The axial strain in the z direction (ends free) equation
Eq. 4
σ_z = v ρ ω² / 4 ( 3 - 2v ) / ( 1 - v ) [ r_i² + r_o² - 2 r² / ( 3 - 2 v) ]

The maximum stress occurs at the inner surface where r = r_i formula
Eq. 5
σ_θ(max) = ρ ω² / 4 ( 3 - 2v ) / ( 1 - v ) [ r_o² + ( ( 1 - 2v ) / ( 3 - 2 v) r_i² ] )

The axial strain in the z direction (ends free) formula
Eq. 6
ε_z = - v ρ ω² / (2 E ) ( r_i² + r_o² )

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 ( r_o² + r_i² + ( 1 + v ) / ( 1 - v) r_o² r_i² / 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)
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:

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