BIZOMAT MAKEOS · Springs
Semi-Elliptic Leaf Spring Design (Leaves for Stress and Deflection)
How many leaves does this semi-elliptic spring need for the load within the allowable stress and deflection, and what rate will it have?
Design of a semi-elliptic (centre-loaded, end-supported) leaf spring of equal-strength graduated leaves: the number of leaves needed to keep the bending stress within the allowable, the number needed to keep the deflection within your limit, the governing count, and the stress, deflection and rate with the leaves you choose (as the leaf spring tool, semi-elliptic case).
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Load at the centre | N | Load applied at the centre clamp (both ends supported). |
| Span between the ends | mm | Distance between the end supports (eyes or slippers). |
| Leaf width | mm | Width of each leaf. |
| Leaf thickness | mm | Thickness of each leaf. |
| Elastic modulus | GPa | Elastic modulus of the spring steel. |
| Allowable bending stress | MPa | Allowable bending stress for the duty (your value). |
| Largest deflection allowed | mm | Largest centre deflection you allow (for clearance or ride). |
| Leaves chosen | Number of leaves you intend to use. |
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: Load at the centre
- What it is
- The total load applied at the centre clamp of the spring, with the two ends resting on their eyes or slippers.
- 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 5000 N as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Stress and deflection both rise in direct proportion.
- Get it right
- This is the whole load at the centre, not the reaction at each end, which is half of it.
What this means: Span between the ends
- What it is
- The distance between the two end supports, eye centre to eye centre, measured straight across the spring.
- 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 1000 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Deflection rises with the cube of span; stress in direct proportion.
- Get it right
- Measure the straight span, not the length along the curved leaf.
What this means: Leaf width
- What it is
- The width of each leaf, measured across the spring.
- 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 60 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Stress and deflection both fall in direct proportion to width.
- Get it right
- All leaves are assumed to be the same width.
What this means: Leaf thickness
- What it is
- The thickness of each leaf, all leaves being taken as equal.
- 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 8 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Stress falls with the square and deflection with the cube of thickness, so small changes matter.
- Get it right
- Measure a single leaf, not the whole pack.
What this means: Elastic modulus
- What it is
- The elastic (Young’s) modulus of the spring steel, which sets how far the leaves bend under load.
- Units and format
- Enter a number in GPa: gigapascals (1 GPa = 1000 MPa). The form accepts 0.000001 GPa to 300 GPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 207 GPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A higher modulus lowers the deflection; it does not change the stress.
- Get it right
- Enter it in GPa, not MPa; take it from the steel data sheet.
What this means: Allowable bending stress
- What it is
- The highest bending stress you accept in the leaves under this load, your design value for the steel and duty.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 0.000001 MPa to 3000 MPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 700 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Choose it for the duty: a static value is too high for a spring that cycles in fatigue.
What this means: Largest deflection allowed
- What it is
- The largest deflection of the spring centre you allow under this load, set by clearance, ride or travel limits.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 2000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 80 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Measure the deflection at the centre, relative to the ends.
What this means: Leaves chosen
- What it is
- The number of leaves you plan to use. The tool reports the stress, deflection and rate for this count beside the number needed.
- Units and format
- Enter a plain number with no unit. The form accepts 1 to 50. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 6 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- More leaves lower stress and deflection in direct proportion.
- Get it right
- Count all the leaves in the pack, including the main leaf.
Method and formulas
- Semi-elliptic, graduated leaves: σ = 3FL ÷ (2 n b t²); δ = 3FL³ ÷ (8 n E b t³) (as the leaf spring tool).
- Leaves for stress n = 3FL ÷ (2 b t² σ_allow); for deflection n = 3FL³ ÷ (8 E b t³ δ_max); round the larger up.
- Rate = F ÷ δ.
- Example: 5000 N on a 1000 mm span needs 4 leaves; 6 leaves run at 325.52 MPa and deflect 49.143 mm.
Limitations
- Equal-strength (graduated) leaves without inter-leaf friction; real springs are stiffer and show hysteresis.
- Static stress; fatigue and the centre bolt hole need their own checks.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Size a leaf spring for a trailer, cart or machine mount
- Check a spring for a higher load
- Trade leaf thickness against leaf count
Who it is for
- Design engineers
- Vehicle engineers
- Maintenance engineers
What you get
- Leaves needed for stress and for deflection
- Stress, deflection and rate with the leaves chosen
Interpreting the result
- Thicker leaves cut the stress-limited count fast, but the spring gets much stiffer (thickness cubed).
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 344–345. 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.
Useful alongside · BIZOMAT original book
Torsion bars and flat springs
What it covers: How to size a torsion bar for stress and twist and balance a heavy panel with it, and how to check flat cantilever springs, with their limits at the ends and in large deflection.
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Open the Semi-Elliptic Leaf Spring Design (Leaves for Stress and Deflection) calculator