Understanding and Calculating Radial and Hoop Stresses in a Rotating Disc

Introduction to Rotating Disc Stresses

When a solid disc rotates at high speed, centrifugal forces act on its mass elements, inducing stresses within the disc. These stresses are primarily radial and hoop stresses. This article explains how to determine the equations for these stresses, derive the necessary formulas, and apply them to a specific scenario.

Derivation of Equations for Radial and Hoop Stresses

To derive the equations for radial and hoop stresses, we consider a small element of the disc. By applying equilibrium conditions and considering the centrifugal forces acting on the element, the following differential equations are obtained:

dσr/dr σr - σθ/r ρω2r

Where: σr: Radial stress σθ: Hoop stress ρ: Density of the material ω: Angular velocity of the disc r: Radial distance from the center

Solving these equations with appropriate boundary conditions σr 0 at r R (the outer radius) gives the following expressions:

σr ρω2R23ν/81-ν - ρω2r23-ν/81-ν

σθ ρω2R23ν/81-ν - ρω2r21ν/81-ν

Here, ν is the Poisson's ratio of the material.

Maximum Stress Calculation

The maximum stress occurs at the center of the disc where both radial and hoop stresses are equal:

σmax ρω2R23ν/81-ν

Calculating Stresses for a Given Disc

For a steel disc with the following parameters:

Diameter D 250 mm Radius R 125 mm 0.125 m Rotational speed N 14000 rpm Density of steel ρ 7850 kg/m3 Poisson's ratio for steel ν 0.3

First, calculate the angular velocity:

ω 2πN/60 2π 14000 / 60 ≈ 1466.08 rad/s

Then calculate the maximum stress:

σmax 7850 kg/m3 1466.08 rad/s2 0.125 m2 30.3/81-0.3

Calculate the numerical value of σmax using a calculator or software.

Note: To calculate the stress at the outside of the disc, simply substitute r R in the equations for σr and σθ.

Application of Concepts to Another Expression

Alternatively, we can use another approach where the stress is expressed in terms of pressures:

Radial Stress σ_r at any radius r:σ_r -P_0r^2/2R^2

Hoop Stress σ_h at any radius r:σ_h ω^2r/2

To calculate the angular velocity from the rotational speed in RPM:

ω 2πN/60

For N 14000 RPM:

ω 2π 14000 / 60 ≈ 1463.76 rad/s

The maximum hoop stress occurs at the outer radius:

R 250 mm / 2 125 mm 0.125 m

Substituting R into the hoop stress formula:

σ_h 0.125^2 1463.76^2 / 2

Calculating this:

σ_h 0.015625 2140463.3 / 2 ≈ 33389.42 / 2 ≈ 16694.71 Pa ≈ 16.69 MPa

The radial stress is zero at the outer edge:

σ_r 0 Pa

Conclusion

By following these steps and using the given values, you can determine the maximum stress and the stress at the outside of the rotating steel disc. This method provides a practical approach to understanding the stresses in a rotating disc and can be applied to various engineering problems.