Belleville Disc Spring

What a Belleville disc spring actually is

Take a flat washer and press a shallow dish into it so it becomes a truncated cone — a ring whose inner edge sits a fraction of a millimetre higher than its outer edge. That coned washer is a Belleville spring, named after the French engineer Julien Belleville who patented the geometry in 1867. Push axially on the high inner edge and the cone tries to flatten; the metal resists, and you have a spring. When you let go, it springs back to its coned free height.

The reason engineers reach for it is brutal force density. A coil spring stores energy by twisting a long wire, so it needs length — to get a big force you need a fat wire and many coils, which means stroke. A disc spring stores energy by bending and circumferentially straining a short, fat annulus, so it delivers an enormous force over a stroke of only a millimetre or two. A single 50 mm outer-diameter DIN 2093 disc can carry several tonnes while compressing less than a millimetre. Pack that into the recess under a bolt head and you have a preload device that no coil spring could fit.

The deflection comes almost entirely from the change in cone angle, not from material strain in the bulk. As the disc flattens, the upper region near the inner edge is squeezed in the circumferential (hoop) direction while the lower region near the outer edge is stretched. Those hoop stresses, not simple bending, dominate the behaviour — which is why a disc spring's load-deflection curve is non-linear in a way no coil spring is.

The governing equation — Almen and Laszlo

The standard model, published by J. O. Almen and A. Laszlo in 1936 and codified in the German standard DIN 2093, gives the axial force F needed to push a single disc to deflection s:

F = (4E / (1 − v²)) · (t⁴ / (K1 · De²)) · (s/t) · [ (h0/t − s/t)(h0/t − s/2t) + 1 ]

E   = Young's modulus            (≈ 206 GPa for spring steel)
v   = Poisson's ratio            (≈ 0.30)
t   = disc thickness
h0  = cone height = free height − thickness
De  = outer diameter
s   = axial deflection
K1  = geometry factor, function of diameter ratio δ = De/Di only

Everything outside the square bracket is a constant for a given disc; it sets the scale of the force. The cubic-in-s/t term inside the bracket is what bends the load-deflection curve. The single most important design number falls straight out of that bracket: the ratio of cone height to thickness, h0/t. It alone decides whether the spring stiffens, holds nearly constant force, or snaps through:

h₀/t ratio	Curve shape	Behaviour	Typical use
< 0.4	Nearly straight line	Almost linear, progressive	Drop-in stiff washer, bolt preload
0.4 – 1.3	Gently S-shaped	Degressive — softens past mid-stroke	General-purpose disc springs
≈ 1.4	Flat-topped plateau	Near constant-force over wide stroke	Constant-tension keepers, valves
> 1.4	Hump then dip	Region of negative stiffness (snap-through)	Over-centre switches, snap-action seals

That last row is the most counter-intuitive property in the whole family: a tall enough disc spring can have a stretch where pushing it further requires less force. Past the hump the disc wants to keep going on its own — the basis of snap-action microswitches, over-centre toggle clamps, and the tactile click in a metal-dome keyboard contact.

A worked example — sizing one disc and a stack

Take a common catalogue disc: outer diameter De = 40 mm, inner Di = 20.4 mm, thickness t = 2.0 mm, free height h0 + t = 2.3 mm so cone height h0 = 0.3 mm, made from 51CrV4 spring steel. With h0/t = 0.15 this is a stiff, near-linear disc.

Diameter ratio  δ = De/Di = 40 / 20.4 = 1.96
Geometry factor K1 ≈ 0.69   (from DIN 2093 chart for δ ≈ 2)

At s = 0.75·h0 = 0.225 mm (75% of flat, the usual working limit):
  Force from one disc ≈ 1.5 kN

Series stack of 4 (><><):   same 1.5 kN  at  4 × 0.225 = 0.9 mm travel
Parallel stack of 4 (>>>>): 4 × 1.5 = 6.0 kN  at  0.225 mm travel
Combo 2-parallel × 3-series: ~3.0 kN at ~0.68 mm travel

This is the entire design lever. You pick a disc whose single-disc force and stroke are in the right ballpark, then use stacking to trade force for travel without changing the disc itself. Need the same force but twice the travel? Add discs in series. Need the same travel but double the force? Nest them in parallel. The catalogue gives you a few dozen discs; stacking gives you a near-continuous design space.

Series versus parallel — the core trade

This is the idea the animation is built around, so it is worth stating plainly. Imagine the discs as arrowheads pointing in the load direction.

• Series alternates direction, ><><, like a concertina. Each disc carries the full load, so the force is that of one disc — but each disc adds its own deflection, so travel multiplies. Four in series gives four times the stroke at the same force.
• Parallel nests them the same way, >>>>, like stacked spoons. They share the deflection, so the travel is that of one disc — but their forces add, so force multiplies. Four in parallel gives four times the force at the same stroke.

The catch with parallel stacks is friction. Each nested face rubs against the next as the discs flex, dissipating energy and adding hysteresis — typically 2 to 6 percent per interface. The loading and unloading curves no longer overlap; the stack returns less force than it took to compress. Sometimes that damping is wanted (vibration isolation under presses and forging hammers deliberately uses lossy parallel stacks). More often it is a nuisance that smears the force value and is mitigated by hardened, lubricated, or low-friction-coated contact faces.

The other practical note: a tall series stack is slender and wants to buckle or tip sideways. It must be guided — threaded onto the bolt shank, or run inside a sleeve, with diametral clearance of roughly the standard's recommended value so the discs can flex without jamming.

Why disc springs hold preload through thermal cycling

The classic reason to slip Belleville washers under a bolt head is to keep the joint tight when the clamped parts change length. A bolt at full preload is an extremely stiff spring — an M16 bolt stretches only about 50–80 µm at its design preload. Anything that shortens the grip by a few micrometres — gasket creep, a hot flange cooling down, surface asperities embedding under load — eats a large fraction of that 50 µm and the clamp force collapses.

Add a disc-spring stack in series with the bolt and you have replaced a tiny-travel stiff spring with a longer-travel softer one. The same few micrometres of lost grip length now represent a small fraction of the stack's working stroke, so most of the preload survives. A disc spring tuned to its constant-force plateau (h0/t ≈ 1.4) is the limiting case: over its working stroke the clamp force barely changes at all, no matter how the joint length wanders. This is why disc springs appear under the head studs of high-temperature steam flanges, in bolted electrical busbar joints that thermal-cycle daily, and as the spindle-bearing preload spring in machine tools.

Where disc springs actually show up

• Clutch diaphragm springs. Almost every modern automotive single-plate clutch uses one large Belleville-style diaphragm spring with radial fingers cut into it. Its falling-rate (negative-stiffness) region is exploited so the pedal effort drops once the clutch is disengaged — the spring helps hold itself open.
• Pressure-relief and safety valves. A disc-spring stack sets the cracking pressure of a relief valve and reopens it crisply; the steep, repeatable curve gives sharp set pressures in a compact head.
• Bolted-joint preload keepers. Under head studs of high-temperature flanges, in busbar connections, and anywhere a joint must survive thermal cycling without retorquing.
• Machine-tool spindle and drawbar preload. A tall parallel-series stack inside the spindle pulls the toolholder taper home with several tonnes while occupying only a few centimetres of axial length.
• Brake and clutch packs in transmissions and limited-slip differentials. Disc-spring washers provide the apply preload and take up wear over the pack's life.
• Overload and vibration isolation. Lossy parallel stacks under forging hammers, presses, and rail-track fastening clips (the Pandrol rail clip is itself a sprung-steel relative) absorb shock and accommodate settlement.
• Snap-action and over-centre devices. Metal-dome keyboard contacts, microswitch buttons, and toggle clamps use the negative-rate hump for a positive click and bistable action.

Failure modes and how to design around them

• Fatigue cracking. The dominant failure. The highest tensile stress sits at the lower inner edge (point III in DIN 2093). Cracks nucleate there and grow radially. DIN 2093 gives S–N bands for 10⁴, 5×10⁵, and 2×10⁶ cycles; shot-peening the loaded edges puts the surface into residual compression and pushes fatigue life up by several times.
• Set and relaxation. Over-stressing past the elastic limit, or operating hot, causes permanent loss of free height — the disc “takes a set” and the preload drifts down. Carbon spring steels relax noticeably above ~200 °C; for hot service use Inconel X-750, 17-7 PH, or other heat-resistant alloys.
• Edge fretting and friction-corrosion. In parallel stacks the rubbing faces fret and corrode, raising friction and scatter over time. Lubrication or low-friction coatings help; so does keeping parallel multiplicity low.
• Buckling or tipping of tall series stacks. A slender unguided column of alternating discs falls over sideways under load. Always guide a series stack on a bolt shank or inside a sleeve with the standard's recommended clearance.
• Flattening beyond the working limit. Pushing past ~75 % of flat drives the inner-edge stress up steeply and risks set; designs keep the working deflection inside that band, or use a deliberate hard stop at the “flat” position.

Disc spring versus the alternatives

Property	Belleville disc spring	Helical coil spring	Wave spring
Force per envelope	Very high	Low–medium	Medium
Axial length needed	Very short (mm)	Long	Short
Stroke per element	Very short (0.3–2 mm)	Long	Short–medium
Load–deflection curve	Non-linear, tunable (incl. constant-force, negative-rate)	Linear	Roughly linear
Tunable by stacking	Yes — series ↔ parallel	No (one rate)	By number of turns
Hysteresis / damping	2–6 % per parallel interface	Low	Low
Cost	Medium	Low	Medium
Best at	High force, tiny space, preload	Long linear travel, low cost	Short axial gaps, light loads

Common pitfalls when specifying disc springs

• Ignoring friction in parallel stacks. The catalogue single-disc force is frictionless. A four-deep parallel stack can return 15–25 % less than the nominal force on unloading. Budget for it or keep parallel multiplicity low.
• Forgetting to guide a series stack. Unguided slender stacks tip and lose load. Specify a guide bolt or sleeve and the right diametral clearance.
• Designing to the flat point. The curve and stress get nasty near flat. Work to ~75 % of flat and leave margin against set.
• Mixing up the h₀/t you actually got. Tolerances on thickness and free height shift h₀/t and therefore the whole curve shape; near the constant-force ratio (~1.4) the sensitivity is highest. Tighten the tolerance band where curve shape matters.
• Using carbon spring steel hot. Above ~200 °C carbon spring steels relax; specify a heat-resistant alloy or expect preload to drift.
• Loading the wrong faces. The load must come onto the inner top edge and outer bottom edge; reversing the support points changes the geometry and the stress distribution entirely.