# STRESSES IN ROTATING RINGS (ANNULAR DISKS)

Stresses in Rotating Rings Calculator was developed to calculate radial / tangential stresses (inertia stresses), the change of the inner and outer radiuses of a ring which rotates about its own axis with a uniform angular velocity. The calculator is valid for homogeneous annular disk of uniform thickness. It has been assumed that the effect of deformation could be ignored and the rotating ring is rigid. Rotating elements, such as flywheels and blowers, can be simplified as a rotating ring and stress calculation can be done.The formulas used for stress calculations are given in the List of Equations section.

If there are radial pressures or pulls distributed uniformly along either the inner or outer perimeter of the disk, these stresses shall be superimposed upon the inertia stresses calculated here. Stresses, moments, and shears due to balanced forces (including gravity) shall also be superimposed upon calculated values in here.

### Rotating Disk Stress Calculator:

 INPUT PARAMETERS Parameter Value Rotation speed [w] rpm Shaft outer radius [R] mm cm m inch ft Shaft inner radius [R0] Radius to the stress element under consideration [r] Density [p] kg/m^3 lbf/in^3 Poisson's Ratio [ν] --- Elastic Modulus  [E] GPa ksi

Note: Use dot "." as decimal separator.

 RESULTS Parameter Value Radial Tensile Stress  (at any point with a distance r from the center) [σr] --- MPa psi Tangential Tensile Stress (at any point with a distance r from the center) [σt] --- Maximum Radial Stress [(σr)max] --- Maximum Tangential Stress [(σt)max] --- Dimensional Change in the Outer Radius [∆R] --- mm cm m inch ft Dimensional Change in the Inner Radius [∆R0] ---

### List of Equations:

 Equation Radial tensile inertia stress [σr] $${ \sigma }_{ r }=\frac { 3+\upsilon }{ 8 } \cdot \rho \cdot { w }^{ 2 }\cdot ({ R }^{ 2 }+{ R }_{ 0 }^{ 2 }-\frac { { R }^{ 2 }{ R }_{ 0 }^{ 2 } }{ { r }^{ 2 } } -{ r }^{ 2 })$$ Tangential tensile inertia stress [σt] $${ \sigma }_{ t }=\frac { \rho \cdot { w }^{ 2 } }{ 8 } \cdot \left[ (3+\upsilon )\cdot ({ R }^{ 2 }+{ R }_{ 0 }^{ 2 }+\frac { { R }^{ 2 }{ R }_{ 0 }^{ 2 } }{ { r }^{ 2 } } )-(1+3\upsilon )\cdot { r }^{ 2 } \right]$$ Maximum radial stress [(σr)max] $${ { (\sigma }_{ r }) }_{ max }=\frac { 3+\upsilon }{ 8 } \cdot \rho \cdot { w }^{ 2 }\cdot { (R-{ R }_{ 0 }) }^{ 2 }$$ Maximum tangential stress [(σt)max] $${ { (\sigma }_{ t }) }_{ max }=\frac { \rho \cdot { \omega }^{ 2 } }{ 4 } \cdot \left[ (3+\upsilon )\cdot { R }^{ 2 }+(1-\upsilon )\cdot { R }_{ 0 }^{ 2 } \right]$$ Dimensional change in the outer radius [∆R] $$\Delta R=\frac { \rho \cdot { \omega }^{ 2 }\cdot R }{ 4\cdot E } \cdot \left[ (1-\upsilon )\cdot { R }^{ 2 }+(3+\upsilon )\cdot { R }_{ 0 }^{ 2 } \right]$$ Dimensional change in the inner radius [∆R0] $${ \Delta R }_{ 0 }=\frac { \rho \cdot { \omega }^{ 2 }\cdot { R }_{ 0 } }{ 4\cdot E } \cdot \left[ (3+\upsilon )\cdot { R }^{ 2 }+(1-\upsilon )\cdot { R }_{ 0 }^{ 2 } \right]$$