Adjust outer radius, bore, rotational speed and material to visualize radial stress σr and hoop stress σθ profiles in real time. Burst speed and safety factor are computed automatically.
Parameters
Outer radius R (mm)
mm
Bore radius r (mm, 0=solid)
mm
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
Results
Results
—
σr max (MPa)
—
σθ max (MPa)
—
Burst speed (rpm)
—
Safety factor SF
—
Max u (mm)
Stress Distribution & Section View
Stress
Disk Stress Distribution
Theory & Key Formulas
Solid disk (r=0):
$$\sigma_r = \frac{3+\nu}{8}\rho\omega^2(R^2 - r^2)$$
$$\sigma_\theta = \frac{3+\nu}{8}\rho\omega^2 R^2 - \frac{1+3\nu}{8}\rho\omega^2 r^2$$
Annular disk: Lamé constants A, B solved from boundary conditions σr=0 at r=ri and r=R.
What is Centrifugal Stress in Rotors?
🙋
What exactly is "hoop stress" in a spinning disk? I see it in the simulator results, but why is it so important?
🎓
Basically, hoop stress ($\sigma_\theta$) is the tensile stress trying to pull the material apart circumferentially, like stretching a rubber band. In a centrifuge rotor, it's caused by the inertia of the mass being flung outward. It's the primary cause of failure. For instance, if you spin a flywheel too fast, it doesn't just bend—it can burst apart radially. Try setting the bore radius to zero in the simulator to see a solid disk; you'll notice the hoop stress is highest right at the center!
🙋
Wait, really? The center has the highest stress? I'd think the outer edge, moving fastest, would be worst. And what's the "radial stress" $\sigma_r$ that's also plotted?
🎓
Good intuition, but for a solid disk, the center is indeed the critical point where $\sigma_r$ and $\sigma_\theta$ are equal and maximum. Radial stress is the stress acting outward from the center, like a pressure. It's always zero at the free outer edge. Now, change the "Bore radius" to something like 20 mm to make an annular disk. See the plot change? The maximum hoop stress jumps to the inner edge of the bore! That's a key design lesson: drilling a central hole (for a shaft) actually increases the peak stress.
🙋
That makes sense! So how do engineers know if a design is safe? The simulator shows a "Safety Margin vs. Yield." Is that just a ratio?
🎓
In practice, we compare the maximum von Mises stress (a combined stress measure) to the material's yield stress. The safety margin tells you how much stronger the material is than the current load. The most critical output is the Burst Speed estimate. It uses the fact that stress is proportional to speed squared ($\omega^2$). So, if your current speed gives a max stress of 100 MPa and the material yields at 400 MPa, the burst speed is $\sqrt{400/100} = 2$ times your current speed. Try doubling the "Speed N" slider and watch how the stress quadruples and the safety margin plummets!
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 $r$.
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
Setting the inner diameter to zero calculates the disk as a solid disk. In this case, at the center (r=0), the radial stress σr and circumferential stress σθ become equal, and the maximum stress occurs. This also affects the burst speed determination, so please set it correctly according to the actual shape.
The safety factor is calculated by dividing the tensile strength of the material (input material properties) by the maximum principal stress obtained from the analysis (usually the maximum circumferential stress). A value below 1 indicates a risk of failure at that rotational speed.
Currently, the tool allows selection from preset materials (e.g., aluminum, steel, titanium, etc.). Density, Poisson's ratio, and tensile strength are automatically set. The ability to add custom materials is under consideration for future updates.
Yes, it does. By setting the inner diameter to a value greater than 0, you can calculate the stress distribution for a hollow disk. You can observe in real time that as the inner diameter increases, the stress concentration at the inner circumference tends to be alleviated.
Real-World Applications
Centrifuge Rotors in Biotech & Chemistry: High-speed centrifuges separate particles by density. A rotor failure at 50,000+ RPM is catastrophic. Engineers use this exact analysis to select materials like titanium alloys and to determine safe operational speeds well below the burst speed, often incorporating a large safety factor.
Flywheel Energy Storage: Modern flywheels spin in a vacuum on magnetic bearings to store kinetic energy. Maximizing rotational speed is key to energy density, but it's directly limited by the tensile strength of the composite rotor. Stress analysis ensures the rotor operates within its fatigue limits for thousands of cycles.
Turbine & Compressor Disks in Jet Engines: The disks that hold turbine blades spin at extreme speeds under high temperatures. Centrifugal stress analysis is combined with thermal stress models. The bore area is critical and is often reinforced, and precise analysis prevents costly and dangerous "disk burst" failures.
Automotive Brake Disks (Rotors): While primarily designed for thermal loads, brake disks also experience centrifugal stresses during high-speed vehicle operation. Understanding the stress profile helps in designing cooling vanes and ensuring structural integrity, especially in performance racing applications.
Common Misconceptions and Points of Caution
When you start using this tool, there are several pitfalls that are easy to fall into, especially for CAE beginners. First is the misconception that "as long as the stress is below the material's yield strength, it's safe." While burst speed is indeed calculated from the yield stress, fatigue failure is a major concern in real-world applications. For instance, a centrifuge rotor spinning at 100,000 rpm can fail due to crack growth from cyclic loading, even if the maximum stress is below 50% of the yield strength. After checking the safety factor with this tool, you must next consider fatigue life.
Next, unit errors in parameter input. This is extremely common. The tool performs calculations internally using SI units (m, kg, rad/s), but design drawings typically use mm for dimensions and rpm for rotational speed. For example, if you inadvertently input an outer diameter of 100mm as "100" (instead of the correct 0.1), the calculated stress will be off by a factor of 10,000, showing a deceptively safe result that could lead to a major failure. Always perform a sanity check after inputting values to ensure the resulting stress is within a reasonable order of magnitude (e.g., tens to hundreds of MPa for steel).
Finally, it's crucial to understand the limitations of the "thin disk" theory. The fundamental equations used by this tool assume a constant and relatively small disk thickness. However, actual turbine disks have significant thickness variations at the hub and blade attachment regions. For such complex geometries, the tool's results are only a first-order approximation. After using the tool to check the stress concentration at the inner bore, the standard practice is to use that value as a reference and perform shape optimization with more detailed 3D FEM analysis.