BIZOMAT MAKEOS · Springs
Flat and Leaf Spring
How much does this flat or leaf spring deflect, and how highly is it stressed?
Deflection, rate and bending stress of a single flat cantilever spring, a graduated multi-leaf cantilever, or a semi-elliptic leaf spring loaded at the centre.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Spring type | choice | Spring arrangement. |
| Load (total load for semi-elliptic) | N | Load at the free end (cantilever) or total centre load (semi-elliptic). |
| Length (span for semi-elliptic) | mm | Length from the clamp to the load, or span between eyes (semi-elliptic). |
| Width | mm | Width of the strip or leaves. |
| Thickness (each leaf) | mm | Thickness of each leaf. |
| Number of leaves | leaves | Number of leaves (1 for a single strip). |
| Elastic modulus | GPa | Elastic modulus of the spring material. |
| Allowable bending stress | MPa | Allowable bending stress for the spring material and duty. |
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: Spring type
- What it is
- The spring arrangement: a single flat strip clamped at one end, a graduated stack of leaves as a cantilever, or a semi-elliptic spring supported at its eyes and loaded at the centre.
- Units and format
- Choose one of 3 options from the list: Single flat cantilever (rectangular); Multi-leaf cantilever (graduated leaves); Semi-elliptic, load at centre.
- Example
- The form opens with “Single flat cantilever (rectangular)” selected as a worked example. Choose the option that matches your case.
What this means: Load (total load for semi-elliptic)
- What it is
- For a cantilever, the force at the free end; for a semi-elliptic spring, the total load at the centre between the two eyes.
- Units and format
- Enter a number in N: newtons. The form accepts 0.000001 N to 100000000 N. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 15 N as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Deflection and stress rise in proportion to load.
- Get it right
- For the semi-elliptic case enter the whole centre load, not half of it.
What this means: Length (span for semi-elliptic)
- What it is
- For a cantilever, the length from the clamp to the load point; for a semi-elliptic spring, the span between the eyes.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100000 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
- Deflection rises with the cube of length and stress in direct proportion.
- Get it right
- Measure from the edge of the clamp, not from the end of the strip under the clamp.
What this means: Width
- What it is
- The width of the strip or of each leaf, measured across the spring at right angles to its length.
- 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 10 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 inverse proportion to width.
What this means: Thickness (each leaf)
- What it is
- The thickness of the strip, or of each individual leaf in a stack, measured in the direction the spring bends.
- 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 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Deflection falls with the cube of thickness and stress with its square, so small changes matter.
- Get it right
- Enter one leaf’s thickness, not the stack height.
What this means: Number of leaves
- What it is
- The number of full leaves stacked to share the load in a multi-leaf or semi-elliptic spring.
- Units and format
- Enter a number in leaves: leaves. The form accepts 1 leaves to 50 leaves. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1 leaves as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- More leaves reduce deflection and stress in inverse proportion.
- Get it right
- Used only for multi-leaf and semi-elliptic springs; the single flat cantilever ignores it.
What this means: Elastic modulus
- What it is
- The elastic (Young’s) modulus of the spring material, which sets how much the leaves deflect.
- 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 207 GPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Deflection falls in inverse proportion to the modulus; stress does not change.
- Get it right
- Enter gigapascals, not megapascals.
What this means: Allowable bending stress
- What it is
- The highest bending stress the spring material may carry for its duty, from the spring maker or material data; the calculated stress is compared with it.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 0.000001 MPa to 100000 MPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 900 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- A spring that cycles needs a fatigue-based allowable, lower than a static one.
Method and formulas
- Single: σ = 6FL ÷ (b t²); δ = 4FL³ ÷ (E b t³).
- Multi-leaf (graduated): σ = 6FL ÷ (n b t²); δ = 6FL³ ÷ (n E b t³).
- Semi-elliptic (total load W, span L): σ = 3WL ÷ (2 n b t²); δ = 3WL³ ÷ (8 n E b t³).
- Example: 15 N on a 60 × 10 × 1 mm flat spring deflects 6.2609 mm with σ = 540 MPa.
Limitations
- Small deflections; large deflections of slender strips need large-deflection theory.
- Friction between leaves and the effect of clamp bolts are ignored.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Design flat spring clips, contacts and detents
- Check vehicle or trailer leaf springs
- Choose strip thickness for a rate
Who it is for
- Design engineers
- Spring buyers
- Maintenance engineers
What you get
- Deflection
- Spring rate
- Maximum bending stress
Interpreting the result
- For flat springs, stress at the clamp governs; avoid sharp clamp edges.
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.
Method explained here · 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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