**Related Resources: calculators**

### Solid Thin Uniform Disk Rotating External Pressure Stress

Engineering Analysis

Strength and Mechanics of Materials Menu

Solid Thin Uniform Disk Rotating at *ω* rad/s Under External Pressure *p _{o}* Stress Equations and Calculator

Preview: Solid Thin Uniform Disk with External Pressure Rotating Stress Calculator

The stresses are

The radial stress at any radius r Forumula

Eq. 1

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

The tangential stress at any radius r

Eq. 2

*σ _{θ} = -p_{o} + ρ ω^{2} ( ( 3 + v ) / 8 ) ( r_{o}^{2} - ( 1 + 3 v ) / ( ( 3 + v ) ) *

The maximum radial stress at r = 0

Eq. 3

*σ _{r(max)} =*

*- ρ + ρ ω*

^{2}( 3 + v ) / 8 r_{o}^{2}The maximum radial stress at *r = r _{o}*

Eq. 4

*σ*

_{r}=*-p*

_{o}The maximum tangential stress at *r = 0*

Eq. 5

*σ _{θ(max)} = σ_{r(max)}*

The displacement u at any radius r

Eq. 6

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

Figure 1 Rotating disk of uniform thickness under external pressure.

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})

*p*= pressure, Pa [psi (lb/in

_{o}^{2})]

*ω*= angular velocity of the ring, rad/s

*v*= Poisson's ratio

u = displacement, m (in)

Related:

- Hollow Cylinder Rotating Stress Equations and Calculator
- 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
- 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