Disc/Belleville Spring Calculator
Estimate tensile stress, axial deflection, and stiffness for a thin annular disc spring under axial force.
Engineering screening tool—verify inputs and results independently. Use at your own risk.
Spring inputs
Use consistent positive dimensions. The inner diameter must be smaller than the outer diameter.
Calculated response
Stress is the model’s nominal tensile measure. Compare it with an independently selected allowable stress that accounts for material, fatigue, temperature, manufacturing, and safety factor.
How to use
- Enter the outer diameter Dₒ and inner diameter Dᵢ in mm, with 0 < Dᵢ < Dₒ.
- Enter uniform thickness t in mm, Young’s modulus E in GPa, and positive axial compressive force F in N.
- Select Calculate, or edit a field for an immediate update. Select Reset to restore the 15 mm, 8 mm, 1 mm, 210 GPa, and 100 N example.
- Read nominal tensile stress, axial deflection, stiffness, and geometry factor. Outputs follow the displayed units and stress/deflection/stiffness are rounded to two decimals.
- Correct any warning before using a result. Do not use this simplified linear-elastic model for snap-through, large deflection, cyclic fatigue life, plasticity, contact-edge stress, stacked springs, or safety-critical final design.
Worked example
1. For Dₒ = 15 mm, Dᵢ = 8 mm, t = 1 mm, E = 210 GPa, and F = 100 N, the calculator gives q = 0.6444, σ = 64.44 MPa, f = 0.02 mm, and k = 5743.59 kN/m. 2. The values indicate a very stiff, small-deflection response; the nominal stress still requires an application-specific allowable-stress and fatigue check.
Equations used
Radius: rₒ = Dₒ/2
Geometry factor: q = 1 − (2/3)(Dᵢ/Dₒ)
Nominal tensile stress: σ = Fq/t²
Axial deflection: f = 0.65rₒ²σ/(Etq)
Secant stiffness: k = F/f
Dₒ, Dᵢ, t, rₒ, and f use compatible length units; F is force; E and σ use compatible stress units; q is dimensionless; k is force per length. The calculator converts mm, GPa, and N internally and reports MPa, mm, and kN/m. Force is entered as a positive compression magnitude, so no algebraic sign is applied.
Theory and method
The calculation treats the component as a thin annular disc in linear elasticity. The diameter ratio enters the nominal stress through q. Substitution of the stress equation into the deflection equation cancels q, giving f = 0.65rₒ²F/(Et³); stiffness therefore depends strongly on thickness and radius in this screening relation.
Assumptions and limits: homogeneous isotropic material, uniform thickness, concentric axial loading, small deformation, and elastic response. Cone height, Poisson’s ratio, edge contact, residual stress, friction, material nonlinearity, and spring stacking are not represented. The relation is not a DIN EN 16983/16984 design calculation and must not replace detailed Belleville-spring stress, fatigue, relaxation, or stability assessment.
Reference: A. van Beek, Advanced Engineering Design: Lifetime Performance and Reliability, spring calculation collection; equations reproduced above. For standardized production spring design, consult DIN EN 16983 and DIN EN 16984 and the material supplier’s fatigue data.
Continue in TriboSolver to refine material, loading, contact, or nonlinear behavior with a higher-fidelity engineering workflow.
