BIZOMAT MAKEOS · Springs
Disc Spring Stack Design
How many disc springs in series and in parallel give this force over this stroke?
Design of a disc (Belleville) spring stack from a single disc’s dimensions: the number of discs in series so that each deflects no more than a chosen share of its cone height over the stroke, the number in parallel to reach the force (Almen–László characteristic, as the disc spring tool), the stack force, free and solid heights, and the top inner-edge stress against an allowable.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Outside diameter | mm | Outside diameter of one disc. |
| Inside diameter | mm | Inside diameter. |
| Thickness | mm | Disc thickness. |
| Cone height h₀ (unloaded, inside) | mm | Unloaded cone height of one disc (overall height minus thickness). |
| Elastic modulus | GPa | Elastic modulus. |
| Poisson’s ratio | Poisson’s ratio. | |
| Force required at the working stroke | N | Force the stack must give at the working stroke. |
| Working stroke of the stack | mm | Total compression of the stack at the working point. |
| Largest deflection per disc (share of h₀) | % | Largest share of cone height each disc may deflect (75% keeps clear of flat). |
| Allowable stress at the top inner edge | MPa | Allowable static stress at the top inner edge (from the spring maker). |
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
- Outside diameter of one disc spring, from the spring maker’s catalogue.
- 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 50 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Inside diameter
- What it is
- Diameter of the central hole of one disc, which fits over the guide rod.
- 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 25.4 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Thickness
- What it is
- Thickness of the steel one disc is made from, from the catalogue or measured with a micrometer.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100 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.
What this means: Cone height h₀ (unloaded, inside)
- What it is
- How far the inner edge stands above the outer edge on one unloaded disc: overall free height minus thickness. It is the travel available before the disc goes flat.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100 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
- Not the overall height: subtract the thickness from the free height.
What this means: Elastic modulus
- What it is
- Elastic (Young’s) modulus of the spring steel, from the spring maker or material data.
- Units and format
- Enter a number in GPa: gigapascals (1 GPa = 1000 MPa). The form accepts 0.000001 GPa to 500 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 per disc and edge stress are both proportional to the modulus.
- Get it right
- Enter GPa, not MPa.
What this means: Poisson’s ratio
- What it is
- Poisson’s ratio of the spring material: how much it contracts sideways per unit of stretch, from the 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.
What this means: Force required at the working stroke
- What it is
- Force the whole stack must push with when compressed by the working stroke.
- Units and format
- Enter a number in N: newtons. The form accepts 0.000001 N to 10000000 N. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 12000 N as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- More force needs more discs in parallel in each group.
- Get it right
- Total force from the stack, not per disc.
What this means: Working stroke of the stack
- What it is
- Total compression of the whole stack from its free length to the working point.
- 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 3 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A longer stroke needs more groups in series.
What this means: Largest deflection per disc (share of h₀)
- What it is
- Largest deflection each disc may take, as a share of its cone height h₀. It decides how many groups are stacked in series.
- Units and format
- Enter a number in %: percent: the number is read as a percentage, so type 8 for 8 %. The form accepts 10% to 100%. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 75% as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A lower share means more groups in series and a longer stack.
- Get it right
- 100 % means the disc is pressed flat.
What this means: Allowable stress at the top inner edge
- What it is
- Stress at the top inner edge of the disc you will accept, from the disc spring maker’s data for your duty.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 0.000001 MPa to 5000 MPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2400 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Make sure the maker’s limit is for the same point on the disc (top inner edge) and the same duty, static or fatigue.
Method and formulas
- Series count n_s = ⌈stroke ÷ (share × h₀)⌉; deflection per disc s = stroke ÷ n_s.
- Force per disc F(s) by Almen–László: F = 4E ÷ (1 − ν²) × t⁴ ÷ (K₁ D_e²) × (s ÷ t) [(h₀ ÷ t − s ÷ t)(h₀ ÷ t − s ÷ 2t) + 1].
- Parallel count n_p = ⌈F ÷ F(s)⌉; stack force = n_p F(s).
- Unloaded stack height = n_s (h₀ + n_p t); solid height = n_s n_p t.
- Example: 12000 N over 3 mm needs 3 groups in series of 3 discs in parallel.
Limitations
- Friction between nested discs raises force when compressing and lowers it when releasing (not included).
- Long series stacks need guiding to stay aligned.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Design a disc spring pack for a clamp, brake or bolted joint
- Find how many discs a stroke and force need
- Check stack height for the space available
Who it is for
- Design engineers
- Spring buyers
- Machine designers
What you get
- Discs in series and parallel
- Stack force and deflection per disc
- Unloaded height, solid height and stress
Interpreting the result
- Parallel discs multiply force; series discs multiply stroke.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 350–353. Topic reference; no source tables or recommended settings are reproduced.
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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