Uses a practical engineering approximation for Belleville (disk) spring behavior. Units: N, mm, N/mm.
What is a disk spring?
A disk spring (also called a Belleville washer) is a conical spring washer that stores energy through elastic deformation. Compared with a coil spring, a disk spring can generate very high force in a short stroke, making it useful in bolted joints, clutches, valves, shock loading systems, and preload assemblies.
How this disk spring calculator works
This page estimates force, spring rate, and energy for a single disk spring and stacked configurations. The stack model supports:
- Series stacking (increased travel, lower stiffness)
- Parallel stacking (increased force, higher stiffness)
- Mixed stacks using series and parallel counts together
Single-spring base stiffness:
k0 = [4 E t³] / [3 (1 - ν²) (Do² - Di²)]Single-spring force at deflection
s:F = k0 · s · (1 + 2h0/t - s/t)Stack relations:
sper = sstack / Ns,
Fstack = Fsingle · Np,
kstack ≈ (dF/ds)single · Np/Ns
Input guidance
Geometry
Enter outer diameter, inner diameter, thickness, and free cone height carefully.
The most common input mistake is reversing inner and outer diameters.
Make sure Do > Di.
Material properties
For many steel disk springs, Young's modulus is around 206 GPa and Poisson's ratio is around 0.30. If you use stainless steel or specialty alloys, update those values based on your material datasheet.
Stack layout
Use series count to represent springs nested in opposite directions for more travel. Use parallel count to represent springs nested in the same direction for more force. This is a fast way to compare design options before detailed finite-element analysis.
Design tips for better results
- Keep operating deflection within the recommended working range from your standard or supplier.
- Watch for negative incremental stiffness near over-flattening behavior.
- Use lubrication and proper guide surfaces to reduce friction and hysteresis effects.
- Check fatigue life for cyclic loading applications.
- Validate critical designs using DIN 2092/2093 data, test rigs, or FEA.
Why engineers use disk spring stacks
Disk spring stacks are compact and tunable. You can tailor force-deflection behavior by changing orientation and count, which makes them popular in precision preload systems and heavy-duty clamping applications. They are especially useful where installation space is limited but a high clamp force is required.
Frequently asked questions
Is this calculator suitable for final certification?
Treat this as a preliminary design calculator. For safety-critical products, always verify with relevant standards, manufacturer curves, and physical testing.
What if my result seems too high?
Double-check units and geometry first. Then check stack arrangement and deflection input. Small geometry changes can strongly affect disk spring force.