Calculation
σ0 = (3+ν) · ρ · ω² · R² / 8
Centrifugal loading stresses a solid disk even without applied torque. Radial and hoop stresses coincide and are largest at its centre.
Spannung im Mittelpunkt einer freien rotierenden Vollscheibe · σ0 = (3+ν)ρω²R²/8
Calculate Centre Stress in a Rotating Solid Disk
Centrifugal loading stresses a solid disk even without applied torque. Radial and hoop stresses coincide and are largest at its centre.
MINTSI
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Result
Select a target and calculate.
Understand the inputs
- Centre stress σ0 — Maximum radial and circumferential tensile stress of the ideal solid disk at its centre; check material limits and other loads before design release.
- Material density ρ — Mass per disk volume from material data; steel is roughly 7850 kg/m³. Use density of the actual material.
- Rotational speed n — Revolutions per minute at the operating point; check maximum operating and possible overspeed separately. Internally n is converted to angular velocity ω.
- Outer radius R — Distance from rotation axis to the free disk edge; halve outer diameter. The 0.2 m value is an arithmetic example.
- Poisson ratio ν — Dimensionless transverse strain per longitudinal strain of the material. Obtain from material data; for many metals ν is roughly 0.3.
Example
Steel disk with ρ = 7850 kg/m³, n = 3000 rpm, R = 0.2 m and ν = 0.3: ω = 314.16 rad/s and σ0 = 12.78 MPa.
Assumptions and limits
Thin solid disk of constant thickness with free rim, no bore, homogeneous isotropic linear elastic material and steady rotation. Hubs, notches, bores, interference fits, temperature and fatigue are excluded; not a complete strength verification.
Understand Stress at the centre of a rotating solid disk
Even without drive torque, a fast-spinning disk is stressed by its own mass. This calculator estimates maximum stress at the centre of an ideal solid disk.
What does this quantity describe?
Each piece of mass requires inward force to move in a circle, creating internal tension. In a thin constant-thickness solid disk with a free rim and no bore, radial and hoop stress coincide at the centre: σ0 = (3+ν)ρω²R²/8. Here ρ is density, ν Poisson ratio, ω angular speed and R outer radius. The calculator converts entered speed n to ω = 2πn/60 when n is in revolutions per minute.
Formula and variables
σ0 = (3+ν) · ρ · ω² · R² / 8
Angular speed: ω = 2πn/60 for n in rpmCentre stress: σr(0) = σt(0) = (3+ν)ρω²R²/8
| Symbol / input | Meaning |
|---|---|
| Centre stress σ0 | Maximum radial and circumferential tensile stress of the ideal solid disk at its centre; check material limits and other loads before design release. |
| Material density ρ | Mass per disk volume from material data; steel is roughly 7850 kg/m³. Use density of the actual material. |
| Rotational speed n | Revolutions per minute at the operating point; check maximum operating and possible overspeed separately. Internally n is converted to angular velocity ω. |
| Outer radius R | Distance from rotation axis to the free disk edge; halve outer diameter. The 0.2 m value is an arithmetic example. |
| Poisson ratio ν | Dimensionless transverse strain per longitudinal strain of the material. Obtain from material data; for many metals ν is roughly 0.3. |
Choose the inputs correctly
Obtain ρ in kg/m³ from material data; steel is roughly 7850 kg/m³. n is actual or assessed rotational speed, such as 3000 rpm. R is outer radius, not diameter. ν is the dimensionless Poisson ratio from material data, often near 0.3 for metals. The simplified input range allows 0 to 0.5.
How to use the calculator
Enter density, speed, radius and Poisson ratio. Check the highest relevant speed, including possible overspeed, separately. Do not treat the stress result as an allowable speed: other loads and material limits belong to a full verification.
Worked example
For a steel solid disk with ρ = 7850 kg/m³, n = 3000 rpm, R = 0.20 m and ν = 0.30, ω = 314.16 rad/s. Hence σ0 = (3.3/8)·7850·314.16²·0.20² = 12.78 MPa.
Understand the result and units
Stress grows with the square of speed and radius. Doubling either multiplies it by four. A bore or fixed hub changes boundary conditions fundamentally and may produce much higher stress.
The calculation uses kg/m³, rad/s, metres and pascals internally. Input rpm is converted to rad/s before squaring; output MPa means million pascals.
Typical applications
Early checks of solid rotating disks, small flywheels and disk rotors without a bore before detailed strength analysis.
Assumptions, limits and common mistakes
Thin homogeneous isotropic linearly elastic solid disk of constant thickness, free rim, without bore, hub, interference fit, notch or external stress. Bending, temperature, fatigue and fracture are excluded. Not a release check for safety-critical rotors.
Common mistake: Do not use diameter as R; doing so quadruples the stress. Revolutions per minute cannot be substituted directly for ω. The model does not cover annular disks with a bore.
Frequently asked questions
What is “Stress at the centre of a rotating solid disk” used for?
Early checks of solid rotating disks, small flywheels and disk rotors without a bore before detailed strength analysis.
Where do the input values come from?
Obtain ρ in kg/m³ from material data; steel is roughly 7850 kg/m³. n is actual or assessed rotational speed, such as 3000 rpm. R is outer radius, not diameter. ν is the dimensionless Poisson ratio from material data, often near 0.3 for metals. The simplified input range allows 0 to 0.5.
What does the result not cover?
Thin homogeneous isotropic linearly elastic solid disk of constant thickness, free rim, without bore, hub, interference fit, notch or external stress. Bending, temperature, fatigue and fracture are excluded. Not a release check for safety-critical rotors.