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Structural Analysis

Centrifuge Rotor Stress Analyzer

Explore the elastic stress field of a homogeneous, constant-thickness thin disk using outer radius, bore radius, speed and material. Yield comparisons are educational; actual burst and safe operating speeds are not calculated.

Parameters
Outer radius R (mm)
mm
Bore radius r (mm, 0=solid)
mm
Target speed N (rpm)
rpm
Young’s modulus E (GPa)
GPa
Poisson’s ratio ν
Density ρ (kg/m³)
kg/m³
Yield stress σy (MPa)
MPa
Presets
Live results
—
Current speed (rpm)
—
Angular velocity ω (rad/s)
—
Maximum hoop stress σθ (MPa)
—
Maximum radial stress σr (MPa)
—
Tip speed vtip (m/s)
—
Rim radial growth u(R) (μm)
—
Hoop-yield threshold speed (rpm)
—
Hoop-yield speed ratio (%)
—
Yield stress ratio SF
Rotating disk (stress color map)
Stress vs. radius (σθ / σr profiles)
Speed vs. stress (yield comparison)
Theory & Key Formulas

$$\begin{aligned}\sigma_\theta^{\,\mathrm{solid}}(0)&=\sigma_r^{\,\mathrm{solid}}(0)\\&=\frac{3+\nu}{8}\rho\,\omega^2 R^2\end{aligned}$$

Maximum stress in a solid disk occurs at the center (σθ=σr). ρ=density, ω=angular velocity [rad/s], R=outer radius [m].

$$\sigma_{ring}=\rho\,\omega^2 r^2=\rho\,v^2$$

Approximate hoop stress in a thin rotating ring (radial thickness much smaller than radius), set by density ρ and tip speed v=ωr; it does not depend on E or ν. It is not the solid or bored disk result.

$$\begin{gathered}\omega_{y,\theta}=\omega\sqrt{\dfrac{\sigma_y}{\sigma_{\theta,max}}}\\SF=\dfrac{\sigma_y}{\sigma_{\theta,max}}\end{gathered}$$

This elastic threshold is where maximum hoop stress equals the input yield stress, not a physical burst speed. SF=σy/σθ,max is not a von Mises or fatigue-based design safety factor.

What is Centrifugal Stress in Rotors?

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How should I enter the dimensions?
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Enter outer radius R and bore radius r in mm. An outer diameter of 240 mm means R=120; a bore diameter of 60 mm means r=30. Conversion to metres is automatic: do not enter 0.12 for a 120 mm radius.
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What does a central hole change?
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With free inner and outer edges, a solid disk has its maximum stress at the centre, while an annular disk has its maximum hoop stress at the bore. At fixed outer radius, density, Poisson ratio and speed, increasing the bore radius increases this peak. Shaft interference and keyways are not modelled.
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Can I safely run at the displayed yield-threshold speed?
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No. It is only the speed where this elastic hoop-stress estimate equals the input yield stress. Because stress scales with speed squared, 100 MPa at a current speed and 400 MPa yield stress gives a threshold twice that speed. This does not predict actual rupture, fatigue life or safe operation.

Physical Model & Key Equations

The governing equations for stress in a rotating disk come from elasticity theory, balancing centrifugal body forces. For a solid disk (no central bore), the radial and hoop stresses vary with the radius $r$.

$$ \sigma_r = \frac{3+\nu}{8}\rho\omega^2(R^2 - r^2) $$ $$ \sigma_\theta = \frac{3+\nu}{8}\rho\omega^2 R^2 - \frac{1+3\nu}{8}\rho\omega^2 r^2 $$

Where:
$\sigma_r, \sigma_\theta$ = Radial & Hoop Stress [Pa]
$\rho$ = Material Density [kg/m³]
$\omega$ = Angular Velocity [rad/s] ($\omega = 2\pi N / 60$)
$R$ = Outer Radius [m]
$r$ = Radial coordinate [m] (0 at center)
$\nu$ = Poisson's Ratio
Note: At the center ($r=0$), $\sigma_r = \sigma_\theta$.

For an annular disk (with a bore of radius $r_i$), the stresses are expressed using Lamé constants A and B, determined by boundary conditions (radial stress = 0 at both inner and outer free surfaces).

$$ \sigma_r = A - \frac{B}{r^2}- \frac{3+\nu}{8}\rho\omega^2 r^2 $$ $$ \sigma_\theta = A + \frac{B}{r^2}- \frac{1+3\nu}{8}\rho\omega^2 r^2 $$

Physical Meaning: The $A \pm B/r^2$ terms represent the static pressure solution, while the terms with $\omega^2$ are the dynamic loading from rotation. The bore creates a stress concentration, making the maximum hoop stress occur at the inner radius, which is often the design-limiting factor.

Frequently Asked Questions

Zero bore radius gives a solid disk. Radial and hoop stress are equal and maximal at its centre. Both geometry inputs are radii in mm, not diameters.
SF=input yield stress/maximum hoop stress. It does not use tensile strength or von Mises stress. SF above 1 does not establish actual burst resistance, fatigue life or safe operation.
Yes. Alongside steel, aluminium and titanium presets, density, Poisson ratio and yield stress can be edited. This free-edge thin-disk stress solution is independent of E; displacement is not calculated by the current tool.
At fixed outer radius, density, Poisson ratio and speed, bore-edge hoop stress increases as bore radius increases. In the thin-ring limit it approaches rho*omega^2*R^2.

Real-World Applications

Rotating disks and flywheels: compare radius, density and speed in a simplified learning model; this does not determine operating limits.

Centrifuge rotors: bowl geometry, specimen loads, shaft contact and fatigue are omitted. Manufacturer speed limits and inspection requirements remain essential.

Turbines and compressors: variable thickness, blade loading, thermal stress and shaft interference require separate analysis.

Composite flywheels: material anisotropy and interlaminar failure are outside this isotropic model.

Common Misconceptions and Points of Caution

Stress below yield is not proof of safety. This is a linear-elastic, homogeneous, isotropic, constant-thickness thin disk with free edges, not a plasticity or fracture model.

Input radii in mm, density in kg/m³ and speed in rpm. An outer diameter of 100 mm requires outer radius 50. SI conversion is automatic.

Fatigue, cracks, corrosion, temperature-dependent strength, shaft contact, keyways and dynamic loads require separate evaluation. Do not infer a universal design safety factor or safe speed from this page.

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How to Use

  1. Choose a material and check density rho(kg/m³), Poisson ratio nu and yield stress(MPa).
  2. Enter outer and bore radii in mm, not diameters. Geometry requires 0≤r<R.
  3. Set target speed N(rpm). During animation, stresses refer to the displayed current speed; check that speed before comparing results.
  4. Compare maximum hoop and radial stress and SF=yield/maximum hoop stress. These values do not certify durability, rupture resistance or safe operation.

Worked Example

Reproducible elastic example, not design approval: steel preset; outer radius 120 mm, bore radius 30 mm, speed 6000 rpm, density 7850 kg/m³, Poisson ratio 0.30 and yield stress 490 MPa. At 6000 rpm the exact hoop peak is 37.3049 MPa at the bore; the radial peak is 10.3547 MPa. SF=13.1350 and the hoop-yield threshold is 21745.35 rpm. Rounded outputs: 37.3 MPa, 10.4 MPa, 13.14 and 21745 rpm. Changing E does not change this stress solution; E only sets the rim radial growth u(R)=R·σθ(R)/E, which is 5.89 μm here. During animation compare against the displayed current speed.

Practical Notes

  1. Stress scales linearly with density and quadratically with speed. Material comparison depends on density, Poisson ratio and yield stress, not E alone.
  2. Keyways, other holes and thickness changes do not have a universal correction percentage; use geometry-specific analysis.
  3. Fatigue life and temperature-dependent strength require actual material data and operating history.
  4. Results are withheld when bore radius is at or above outer radius, a radius is blank, or a radius is outside its input range. Correcting the geometry restores calculation.