Belleville Spring Rate⚠ unverified

Understanding the Parameters

• Disc thickness $t$ — enters as $t^3$, the dominant stiffness lever, exactly as in any bending element. Thicker discs are far stiffer and carry far more load; stacking thin discs instead keeps the rate low.

• Outer diameter $D$ — enters as $1/D^2$: larger-diameter discs are softer. It sets the overall size and, with $d$, the load-bearing annulus.

• Free cone height $h$ — the height of the cone, i.e. the axial travel available before the disc goes flat. The ratio $h/t$ is the key nonlinearity control: for $h/t \lesssim 0.4$ the response is nearly linear; near $h/t \approx 1.4$ the curve flattens (constant force); above $\approx 1.5$ it develops a negative-rate snap-through region. The $(h/t + 1)^2$ grouping is this formula's crude nod to that dependence.

• Inner diameter $d$ — physically important (it sets the diameter ratio and the geometry factor) but absent from this simplified formula.

• Poisson's ratio (0.3) — hard-coded; the $1/(1-\nu^2)$ factor is the plate-stiffening term from plate-bending theory, fixed here at the steel value.

Derivation (Approaching a Proof)

The rigorous model is the Almen–László theory (1936), which treats the disc as a thin annular plate that changes cone angle as it deflects. It gives the axial load $P$ as a cubic function of deflection $s$:

$$P(s) = \frac{E\,s}{(1-\nu^2)\,\alpha\,D^2}\left[(h - s)\left(h - \frac{s}{2}\right)t + t^3\right],$$

where $\alpha$ is a dimensionless geometry factor depending on the diameter ratio $D/d$. Because $P(s)$ is cubic, the tangent rate $\mathrm{d}P/\mathrm{d}s$ varies with deflection — it can be positive, zero, or negative — which is the source of the flat and snap-through behaviours.

The registry formula is the small-deflection linearisation: evaluating the stiffness near $s = 0$ (or using an initial-slope approximation) collapses the cubic to a constant. The $E t^3/[6(1-\nu^2)D^2]$ prefactor is the plate-bending stiffness scale, and the $(h/t + 1)^2$ term is a simplified stand-in for the $h$-dependence, with the diameter-ratio factor $\alpha$ (and thus $d$) dropped. It is adequate for a first estimate in the near-linear regime but must not be used where the $h/t$ ratio drives the disc into its flat or negative-rate region — there the full $P(s)$ curve is required.

Dimensional check. $[k] = \dfrac{\text{Pa}\cdot\text{m}^3}{\text{m}^2} = (\text{N/m}^2)\,\text{m} = \text{N/m}$ (the $(h/t+1)$ and $1-\nu^2$ terms are dimensionless). ✓

History and Development

The conical disc spring is named after the French engineer Julien Belleville, who patented it in 1867. Its rigorous load–deflection theory was developed by J. O. Almen and A. László (1936) and remains the industry standard (DIN 2092/2093). Belleville stacks — arranged in series (more travel) or parallel (more force) — are used wherever high force, compact height, or a tailored/constant-force characteristic is needed: bolted-joint preload, clutch and brake plates, valve and relief mechanisms, and overload protection.

Related Concepts: Helical Spring Rate, Spring Energy, Leaf Spring Rate, Beam Bending Stress, Spring Index

Notes: Linearised small-deflection rate only — a Belleville washer is intrinsically nonlinear (Almen–László); use the full $P(s)$ curve near flat/snap-through ($h/t \gtrsim 1$). Inner diameter $d$ is unused and $\nu = 0.3$ is hard-coded (see registry note). Stack in series for travel, parallel for force.