Leaf Spring Rate⚠ unverified
Mechanical / Springs · Compute the spring rate of a leaf spring
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| w | w | m | 1.0 | Width of a single leaf |
| t | t | m | 1.0 | Thickness of a single leaf |
| L | L | m | 1.0 | Span length of the spring |
| n | n | — | 1.0 | Number of leaves (dimensionless) |
| E | E | Pa | 210000000000.0 | Young's modulus of the leaf material, in pascals (Pa). Default is 210e9 |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | k | N/m | Spring rate, in newtons per metre (N/m) |
The science & history
Understanding the Parameters
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Leaf width $w$ — enters linearly through the beam's second moment of area $I = wt^3/12$. Wider leaves are proportionally stiffer; width is the gentle tuning knob.
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Leaf thickness $t$ — enters as $t^3$, the single most powerful lever: a leaf 25 % thicker is nearly twice as stiff. This cube dependence comes straight from beam bending, where material far from the neutral axis dominates.
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Span length $L$ — enters as $1/L^3$: a longer spring is dramatically softer, which is why long leaf springs give a compliant ride. Halving the span makes the spring eight times stiffer.
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Number of leaves $n$ — the leaves act like beams in parallel, so their rates add: $n$ leaves give $n$ times the stiffness (and share the load, lowering each leaf's stress). Graduated leaf lengths also spread bending stress evenly along the spring.
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Young's modulus $E$ — the leaf bends, so bending stiffness uses $E$ (as for any beam), not the shear modulus.
Derivation (Approaching a Proof)
Model one leaf as a cantilever beam of length $L$ carrying an end load $F$. From Euler–Bernoulli beam theory, the end deflection is
$$\delta = \frac{F L^3}{3 E I}, \qquad I = \frac{w t^3}{12}\ \text{(rectangular section)}.$$
So a single-leaf cantilever rate is
$$k_1 = \frac{F}{\delta} = \frac{3 E I}{L^3} = \frac{3 E (w t^3/12)}{L^3} = \frac{E w t^3}{4 L^3}.$$
The factor $\tfrac{3}{12} = \tfrac14$ produces the coefficient in the formula. With $n$ leaves stacked to act in parallel, the rates add:
$$k = n\,k_1 = \frac{E w t^3 n}{4 L^3}.$$
This treats the stack as $n$ identical parallel cantilevers — an idealisation. Real semi-elliptic leaf springs are more like simply-supported/eye-mounted beams with graduated leaf lengths and inter-leaf friction (which adds hysteretic damping); those refinements change the numeric coefficient and add damping but not the fundamental $w t^3 n/L^3$ scaling.
Dimensional check. $[k] = \dfrac{\text{Pa}\cdot\text{m}\cdot\text{m}^3}{\text{m}^3} = (\text{N/m}^2)\,\text{m} = \text{N/m}$. ✓
History and Development
Leaf (or "laminated") springs are among the oldest engineered springs, used on horse-drawn carriages from the 17th century and standard on automobiles and railway rolling stock into the modern era. Their beam- bending analysis is textbook strength-of-materials; the graduated multi-leaf design (constant-strength beam) and inter-leaf friction damping are covered in Wahl and automotive-suspension references. Parabolic single/few-leaf springs are the modern lightweight evolution.
Related Concepts: Beam Bending Stress, Rectangular Moment of Inertia, Helical Spring Rate, Section Modulus, Spring Energy
Notes: Idealised as $n$ parallel cantilever leaves (hence the $1/4$ coefficient). Real semi-elliptic springs (simply supported, graduated leaves, inter-leaf friction) shift the coefficient and add damping. Uses $E$ (bending). Thickness and span dominate the rate.