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### Hollow Cylinder Rotating Stress Equations and Calculator

Engineering Analysis

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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 ω^{2} / *

*8 * ( ( 3 - 2v ) / ( 1 - v ) ) [ r*

_{i}^{2}+ r_{o}^{2}+ r_{i}^{2}r_{o}^{2}/ r^{2 }- ( ( 1 + 2v ) / ( 3 - 2v ) ) r^{2}]The radial stress at any radius r formula

Eq. 2

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

The axial stress (ends free) at any radius r formula

Eq. 3

*σ _{z} =*

*ρ ω*

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

_{i}^{2}+ r_{o}^{2}- 2r^{2})The axial strain in the z direction (ends free) equation

Eq. 4

*σ _{z} =*

*v ρ ω*

^{2}/ 4 ( 3 - 2v ) / ( 1 - v ) [ r_{i}^{2}+ r_{o}^{2}- 2 r^{2}/ ( 3 - 2 v) ]The maximum stress occurs at the inner surface where r = *r _{i}* formula

Eq. 5

*σ*4 ( 3 - 2v ) / ( 1 - v ) [

_{θ(max)}= ρ ω^{2}/*r*+ ( ( 1 - 2v ) / ( 3 - 2 v)

_{o}^{2}*r*] )

_{i}^{2}The axial strain in the z direction (ends free) formula

Eq. 6

ε_{z} = *- v ρ ω ^{2} / (2 E ) ( r_{i}^{2} + r_{o}^{2} ) *

The displacement u at any radius r of a thin hollow rotating disk formula

Eq. 7

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

Figure 1

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

Related:

- Rotating Solid Cylinder Stress Equations and Calculator
- Rotating Rings Bodies Stress Equations and Calculator
- Cylinder Stress and Deflection Rotation Equation and Calculator
- Critical Speeds of Rotating Shafts and Mass Review
- Counterbalancing Masses in Two or More Planes Formulas and Calculator
- Mass Moment of Inertia Equations
- Critical Speeds of Rotating Shafts with Single Loads Equations and Calculators
- Stresses in Rotating Disks (Annular Rings) of Constant Thickness Equation and Calculator
- Beam Deflection and Stress Calculators Formulas & Structural Analysis

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