BIZOMAT MAKEOS · Springs
Disc (Belleville) Spring Load and Stress
What force and stress does this disc spring, or stack, give at a deflection?
Load, flat load and top inner-edge stress of a disc (Belleville) spring by the Almen–László method, and the force and travel of series and parallel stacks.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Outside diameter | mm | Outside diameter from the catalogue. |
| Inside diameter | mm | Inside diameter from the catalogue. |
| Thickness | mm | Disc thickness. |
| Cone height h₀ (free height − thickness) | mm | Unloaded height l₀ minus thickness t, from the catalogue. |
| Elastic modulus | GPa | Elastic modulus of the spring steel. |
| Poisson’s ratio | Poisson’s ratio (about 0.3). | |
| Deflection of one disc | mm | Deflection of a single disc at the working position (no more than h₀). |
| Discs in series (alternating) | discs | Number of single discs (or parallel sets) stacked in alternating directions. |
| Discs in parallel (nested) | discs | Number of discs nested in the same direction in each set. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Outside diameter
- What it is
- The outside diameter of the disc (Belleville) spring, from the catalogue or drawing.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 10000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 50 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- It must be larger than the inside diameter.
What this means: Inside diameter
- What it is
- The inside (bore) diameter of the disc spring, from the catalogue or drawing.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 10000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 25.4 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- It must be smaller than the outside diameter.
What this means: Thickness
- What it is
- The thickness of the disc spring material.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The force rises steeply with thickness.
- Get it right
- Use the catalogue thickness; for discs with contact flats this model does not apply.
What this means: Cone height h₀ (free height − thickness)
- What it is
- The free cone height of one disc: its unloaded overall height minus the material thickness. It is how far the disc can deflect before it is flat.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1.4 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Do not enter the overall free height; subtract the thickness first.
What this means: Elastic modulus
- What it is
- The elastic modulus (Young’s modulus) of the disc spring steel, from the maker or material data.
- Units and format
- Enter a number in GPa: gigapascals (1 GPa = 1000 MPa). The form accepts 0.000001 GPa to 1000 GPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 206 GPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Force and stress rise in proportion to it.
- Get it right
- Enter GPa, not MPa: 1 GPa is 1000 MPa.
What this means: Poisson’s ratio
- What it is
- Poisson’s ratio of the disc spring material, from the maker or material data.
- Units and format
- Enter a plain number with no unit. The form accepts 0 to 0.5. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.3 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- It is a dimensionless ratio, not a percentage.
What this means: Deflection of one disc
- What it is
- How far one single disc is compressed at the working position, measured from its free height.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 1000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1.05 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Enter the deflection of one disc, not of the whole stack; it cannot exceed the free cone height.
What this means: Discs in series (alternating)
- What it is
- The number of single discs, or nested sets, stacked face to face in alternating directions. Their deflections add.
- Units and format
- Enter a number in discs: discs. The form accepts 1 discs to 1000 discs. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1 discs as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Stack travel rises in proportion; force is unchanged.
- Get it right
- It multiplies the stack travel, not the force.
What this means: Discs in parallel (nested)
- What it is
- The number of discs nested in the same direction within each set. Their forces add.
- Units and format
- Enter a number in discs: discs. The form accepts 1 discs to 100 discs. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1 discs as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The stack force rises in proportion; travel is unchanged.
- Get it right
- Friction between nested discs is ignored, so the real loading force is higher and the unloading force lower.
Method and formulas
- δ = De ÷ Di; K₁ = (1/π)((δ − 1)/δ)² ÷ ((δ + 1)/(δ − 1) − 2/ln δ).
- F = 4E ÷ (1 − ν²) × t⁴ ÷ (K₁De²) × (s/t) × [(h₀/t − s/t)(h₀/t − s/2t) + 1].
- σI = −4E ÷ (1 − ν²) × t² ÷ (K₁De²) × (s/t) × [K₂(h₀/t − s/2t) + K₃], with K₂ = (6/π)((δ − 1)/ln δ − 1)/ln δ and K₃ = (3/π)(δ − 1)/ln δ.
- Stack force = F × parallel count (friction neglected); stack travel = s × series count.
- Example: the default disc gives 4,762.1 N at 1.05 mm deflection and 5,897.9 N at flat.
Limitations
- Discs without contact flats; friction in parallel stacks raises loading force and lowers unloading force.
- Fatigue needs the tensile stresses at the lower edges (σII, σIII) and the maker’s fatigue diagrams.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Size disc stacks for bolted joints, clutches and valves
- Check load at a set deflection
- Compare series and parallel stacking
Who it is for
- Design engineers
- Spring buyers
- Maintenance engineers
What you get
- Force of one disc and of the stack
- Force at flat
- Top inner-edge stress (σOM and σI)
- Stack travel
Interpreting the result
- Keep the working deflection at or below about 75% of h₀ for a predictable, near-linear characteristic.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 350–359. Topic reference; no source tables or recommended settings are reproduced.
- EN 16983:2016 Disc springs — Quality specifications — Dimensions; EN 16984:2016 Disc springs — Calculation
Learn the Concepts Behind This Tool
These BIZOMAT Library readings explain the method this tool uses, with worked examples.
Method explained here · BIZOMAT original book
Disc spring stacks: large forces in short spaces
What it covers: How disc (Belleville) springs carry large forces in little space: the Almen–László characteristic, series and parallel stacking, stack design for force and travel, and stresses.
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Open the Disc (Belleville) Spring Load and Stress calculator