Smart Spring Calculator for Belleville (Disc) Washers

Every Belleville washer is described by its outside diameter (OD), inside diameter (ID), thickness (t), and cone height (h) — the vertical rise from the bottom edge to the top edge measured before any load is applied. The overall height (H) is simply the cone height plus material thickness, H = h + t, and is the free height an assembler would actually measure with calipers.

The single most important shape parameter, though, is the ratio of these last two dimensions:

This one number governs whether the disc behaves like a stiff, nearly linear spring or like a device that holds an almost constant force over a wide range of travel — and, taken too far, whether it becomes bistable.

The second governing ratio compares the outside and inside diameters:

R is what every dimensionless factor in this guide is a function of — M, C₁, C₂, T₁, T₂, and the DIN 2092 K₁–K₄ coefficients (card 04) all depend on R alone, not on the absolute size of the washer. Discs proportioned near R ≈ 2 store the most energy per unit of material, which is why that value keeps coming up as a starting point (card 08).

A disc with a low h/t (roughly under 0.4) produces a load-deflection curve that is nearly a straight line — it acts much like a very stiff flat spring, well suited to applications that just need a large force from a small package and don't care about a flat load curve.

As h/t rises toward 1.41, the curve develops a broad, nearly horizontal plateau — for roughly the middle half of its travel either side of the flat position, load barely changes with deflection. This is the proportion designers reach for whenever the goal is a constant clamping force that shrugs off wear, thermal growth, or assembly tolerance.

Push h/t higher still, past about 2.83, and the curve stops being single-valued: once compressed beyond flat, the disc will not return to its original dome on its own — it has snapped into the opposite dish and needs an external force to pop it back. Unless a design intentionally wants that toggle action, h/t should stay below this threshold.

When a Belleville washer is squeezed between two flat surfaces, the calculated load-deflection curve is only trustworthy across roughly 15% to 85% of the cone height h. Below 15%, seating effects and minor edge imperfections dominate; push past 85% and the load rises faster than the ideal-curve math predicts, because the contact geometry itself starts to shift.

Load has to be introduced uniformly around the full circumference at both the top and bottom edge — a few point contacts instead of a continuous ring will cock the disc and produce stresses and deflections nothing like the design calculation.

The corners where a Belleville washer contacts its mating surfaces matter more than they might appear to. A generously rounded edge removes active material and shortens the effective moment arm, which stiffens the disc and raises its rate. A sharp corner does the opposite for rate but concentrates stress right where fatigue cracks like to start — sharp edges buy a lower rate at the cost of cyclic life. Most designs land on a light break edge as the practical middle ground.

Some designs need a precisely controlled load right at, or even beyond, the flat position — for example a mechanism that must stay compact once fully seated. Ordinary flat-plate mounting cannot do this reliably, because as the disc approaches flat, the contact ring migrates and the effective moment arm collapses in an uncontrolled way. A support that locates the load at a fixed radius near the outer and inner edges, and that adds a positive mechanical stop, is needed instead. With that kind of support, the rate actually increases once the disc passes flat, since the moment arm keeps shrinking as travel continues.

Mounting a Belleville Washer for Travel Past Flat

Some Belleville washers are manufactured with a small flat land ground onto the inner and/or outer edge rather than a knife-sharp corner. This controls exactly where the contact line sits, reduces local bearing stress against the mating part, and makes the effective moment arm more predictable. The trade-off is slightly less active spring material.

Stress is far from uniform across a Belleville washer's cross-section. The highest-magnitude stress is compressive and occurs at the convex top corner near the inside diameter (S_c). Two tensile stresses develop on the concave underside: one at the inner edge (S_T1) and one at the outer edge (S_T2). For discs with h/t above roughly 0.6, S_T2 tends to run higher than S_T1. Cyclic designs should check the stress range at both bottom locations rather than assuming the inner edge always governs. See the "Which Point Governs" chart later on this card for exactly when each one does.

Critical Stress Locations

Each of the three stresses follows the same leading term as the load equation, with its own location-specific bracket:

Before the stresses, the same 1936 Almen and Laszlo analysis gives the load-deflection relationship itself, through a shared leading term and a location-specific bracket:

M, C₁, C₂, T₁, T₂ are dimensionless factors that depend only on the diameter ratio R = OD/ID — Almen and Laszlo published them as charts rather than closed-form expressions. The later DIN 2092/2093 standard refines this same relationship with its own closed-form K₁–K₄ coefficients, which is the form most current disc-spring software actually computes with.

Nesting identical washers all facing the same direction — a series stack — multiplies deflection by the number of washers while the load stays that of a single disc. Nesting washers in alternating pairs, concave against concave and convex against convex — a parallel stack — multiplies load by the washer count while deflection stays that of one disc. Series and parallel groups can be combined to reach a target load and deflection that neither arrangement alone would hit.

Both stacking arrangements exhibit hysteresis from friction between adjacent washer faces — worse in parallel stacks, where more surface area slides against itself, than in series stacks. Lubricating the mating faces reduces this friction, though the resulting energy loss is not purely a downside: it damps vibration passing through the stack.

A stacked assembly must be guided, either over a pin or inside a bore, or the discs will walk sideways under load. Guide surfaces should be hardened to at least HRC 50 to resist wear from the sliding washer edges, and diametral clearance between the washer bore/OD and its guide should run around 1.5% of the relevant diameter — tight enough to control position, loose enough not to bind as the stack deflects.

A single disc spring's curve is linear-to-degressive, never progressive (card 02), and stacking alone doesn't change that shape — it only stretches it, per the four schematics above. Getting a genuinely progressive (rising-rate) response out of disc springs takes splitting the stack into sub-groups that "lock out" one at a time as deflection increases, so the remaining active groups get progressively stiffer. Two ways to build that:

Different maximum force per group. Arrange series groups x, y, and z with the same washer count but different individual load ratings. All three groups deflect together at first (combined curve x+y+z), but group x flattens out (reaches its own load-to-flat) at force F_x and stops contributing further deflection, leaving only y+z active and stiffer. Group y flattens next at F_y, leaving z alone to carry the rest of the travel at its own, steeper rate.

The same lockout effect can also be built with identical washers by varying the number of parallel washers per group instead of the individual washer rating — more washers in a bank means that bank reaches its collective load-to-flat sooner, so it's the bank size rather than the washer rating that staggers the lockouts.

Mechanical stops of different thicknesses. Rather than relying on groups flattening on their own, a stepped stop sleeve inside the guide can physically bottom out each section of the stack in sequence as the assembly compresses, engaging progressively stiffer sections on a schedule set by the sleeve's machined steps rather than by the springs' own load-to-flat points.

Three separate contacts generate friction in a real stack: (a) between the end washers and the compressing surfaces, (b) between adjacent washer faces in a nested/parallel group, and (c) between the washer bore/OD and its guide element. Friction (a)+(b) together raise the loading force and lower the unloading force relative to the frictionless calculation — this is the physical cause of the hysteresis loop shown above and in the chart further down this card:

Use the minus sign while loading, the plus sign while unloading. w_M is the friction coefficient between nested washer faces; w_R is the friction coefficient at the compressing-surface contact.

Series A/B/C follow DIN 2093's geometry classes (roughly linear / moderately degressive / highly degressive, per card 02). Friction at the guide element itself (contact (c) above) isn't reliably predictable from geometry alone — it dominates in long stacks or stacks with many small groups, and grows with taper angle, so a guided long stack should be validated by test rather than by these formulas alone.

Hardened carbon or alloy spring steel, typically finished to about HRC 47–50, gives the highest energy storage per unit volume and is the default choice whenever the environment allows it. When corrosion resistance or a non-magnetic part is required, austenitic stainless (301/302-type) is the common substitute, accepting a lower allowable stress in exchange. For elevated-temperature service or aggressive chemical exposure beyond what stainless can handle, nickel-based alloys such as Inconel X-750 retain their spring properties well past the temperature at which steel starts to relax. Beryllium copper and phosphor bronze are reserved for applications needing electrical conductivity or non-sparking behavior.

International standard DIN 2093 catalogs standard disc-spring diameters, thicknesses, and rated loads, which lets many designs use an off-the-shelf part instead of a custom one. SAE and ASTM references cover the spring-quality steel and stainless strip stock that finished discs are stamped from. Checking the catalog options before committing to a custom geometry can save both tooling cost and lead time.

Specify outside diameter with a minus-only tolerance and inside diameter with a plus-only tolerance — that combination guarantees clearance to the mating bore or pin never gets tighter than intended, no matter where in the tolerance band a given part lands.

Load tolerance should always be called out at a specific test height, not just at free height. As a starting point, washers with h/t below about 0.25 typically hold to around ±15% of rated load, while stiffer washers with h/t above 0.25 can usually be held to about ±10%. Nonferrous materials generally run closer to ±15% regardless of h/t. Tighter diameter or load tolerances are available from most manufacturers, but they add cost and should only be called out where the application genuinely needs them.

DIN 2093 sorts every disc spring into one of three groups by raw thickness t alone, and that group decides both how the washer is made and which fatigue-strength chart applies to it (card 06). Group 3 is the odd one out: it's ground or turned to a reduced contact-surface thickness t′ at the top and bottom edges (visible as the "reduced effective lever arm" in the lower cross-section of the figure), which is the K₄ > 1 case already flagged in card 04. Groups 1 and 2 are geometrically identical to each other, made with a flat top and bottom edge all the way to the corner (K₄ = 1, the case this calculator models) — they're only split apart because thin Group 1 stock is usually blanked or cold-formed while Group 2 is more often turned, and because Group 1's thinness earns it a separate fatigue-strength chart in card 06 despite sharing Group 2's geometry.

A spring-loaded electrical contact assembly must maintain at least 220 N of clamping force across 0.50 mm of expected contact wear over its service life. Outside diameter is limited to 42 mm by the surrounding housing, and the part must resist corrosion, so an austenitic stainless washer is selected.

1. Add margin to the minimum load target: design for roughly 10% above minimum, so 220 × 1.10 ≈ 242 N.
2. Pick the diameter ratio R = OD/ID ≈ 2 — the proportion that stores the most energy per unit of material — giving an inside diameter of about 21 mm.
3. Choose h/t = 1.41 from the curve family, since it holds a nearly constant load across the middle of its travel, matching the need to ride out 0.50 mm of wear without the force drifting much.
4. Read the percent-load-at-deflection curve for h/t = 1.41: at roughly 50% of deflection to flat, load runs at about 88% of the load-at-flat value P_F. That gives P_F ≈ 242 / 0.88 ≈ 275 N.

5. Cross-reference P_F and the 42 mm outside diameter against the compressive-stress chart to estimate S_c, and confirm it sits comfortably below the static allowable for stainless without set removed (95% MTS).
6. Solve the load equation for thickness t using P_F, R, h/t, and the material's elastic modulus and Poisson's ratio.
7. Lay out the working window: the initial preload deflection plus the full 0.50 mm of anticipated wear must both land inside the 15–85% band. If the wear pushes the upper end past 85%, shift the whole window down (for example to 40–85% instead of 50–100%) and repeat steps 4–6 with the new percentages.
8. Once the static case closes, check S_T1 and S_T2 against the modified Goodman diagram if the contact is expected to cycle in service, and finalize dimensions and tolerances per the previous section.

• Material and hardness/temper
• Outside diameter (minus-only tolerance)
• Inside diameter (plus-only tolerance)
• Thickness
• Free (cone) height, h, and h/t ratio
• Flat bearing surface, if any, and its width

• Required load and load tolerance, and the test height it's measured at
• Allowable relaxation (%) over service life
• Required cycle life and reliability target, if cyclic

• Bore diameter the washer must clear, or pin diameter it must fit over
• Stack configuration: number in series, number in parallel, total count
• Maximum operating temperature
• Operating environment (corrosive, wash-down, vacuum, etc.)

A bolted joint with no compliant element has almost no elastic reserve. The bolt itself only stretches a few thousandths of an inch before its clamp load is fully spent. Any shrinkage of the clamped stack, from thermal contraction, gasket creep, or settling, can zero out the clamp force entirely. A Belleville stack sized for a large working deflection, the whole point of the 15–85%-of-h range from card 03, turns that same length change into a small slice of a much bigger elastic range. A shrinking gap then costs only a modest fraction of the preload, not all of it. That is the entire reason disc springs get specified into thermally cycled clamps in the first place.

Within that picture, three separate effects move the preload as temperature swings. Only one of them is permanent.

The bolt and the clamped stack are almost never the same material, so they grow and shrink at different rates. Say the clamped stack's coefficient of thermal expansion (CTE) is higher than the bolt's. The classic case is an aluminum housing clamped by a steel bolt. Heating grows the housing faster than the bolt can follow, and the joint loses clamp deflection. Cooling does the reverse. Work this out for your own stack-up. Sum α·ΔT·L for every component in the load path, then compare the clamped-stack growth to the bolt's own growth over its grip length. Don't assume a direction; it flips depending on which material has the higher CTE.

Every equation in this guide carries E, the material's elastic modulus, as a direct multiplier on load. E falls as temperature rises for essentially every engineering metal. The drop is typically a few percent per hundred degrees for steels, more for some superalloys. So the same deflection produces measurably less force when hot:

This tracks temperature directly, and it fully recovers on cooling. It is not damage, just the material getting softer while hot. Get the real E(T) curve for your specific alloy from the material supplier. Don't assume a generic slope; it varies a lot between, say, carbon steel and a nickel superalloy.

This is the mechanism to actually design around. It's a different animal from both effects above. Held under sustained stress at elevated temperature, the material slowly creeps. The disc's residual elastic strain bleeds off over time, permanently shortening its effective free height. Unlike the modulus effect, relaxation does not reverse when the part cools back down. The lost preload is gone for good. It is also strongly time- and temperature-dependent, roughly exponential in absolute temperature. A clamp that looks fine after a short thermal excursion can still lose significant preload if it dwells at that same temperature for months or years.

"Taking a set" (card 06) is a related but distinct failure. It's a single, discrete overstress event, where a stress excursion crosses the material's yield-equivalent limit. Relaxation, by contrast, proceeds continuously, even at stresses the static allowable tables call safe. Heat drives it, not a single bad excursion. The two blend together at high enough temperature, though. Static allowables in this guide assume roughly room temperature. A design margined only against the room-temperature number can find its effective margin quietly eaten away, as sustained heat both relaxes the part and lowers the stress it can tolerate before actually setting.

• Bias the stack toward the h/t ≈ 1.41 constant-force plateau (card 02). Then, whatever deflection the CTE mismatch steals back, the load barely moves.
• Size in margin at both temperature extremes. Recompute with the hot-temperature E(T) and the cold-temperature E(T), not just the assembly-temperature value, and confirm clamp force stays above the application's minimum at both ends.
• For sustained high temperature, choose a material selected for relaxation resistance at that temperature (card 07). A nickel superalloy like Inconel X-750 or A286 will hold preload at temperatures where plain carbon or alloy steel would relax substantially within the first few hundred hours.
• Consider heat-setting the washers. Pre-relax them at or above the expected service temperature before installation, so the bulk of the relaxation loss happens in a controlled process, not silently over years in the field.

Belleville washers, wave washers, curved washers, finger spring washers, and stacked wave springs are all doing the same basic thing: a thin annular ring, formed out of flat, pushes back axially when compressed. What changes between them is how much force they give you, how much stroke they give you, and how much axial height that stroke costs. Picking the right one is mostly a matter of where your application sits on that trade-off, plus a couple of jobs (electrical contact, coil-spring-like travel) that only one shape really covers.

The Belleville washer, the subject of the rest of this guide, sits at one extreme: for a given radial envelope and axial height, it delivers the most force of the five, at the smallest deflection per washer. Stacking (card 05) trades some of that force back for more stroke, but a single disc is still the highest-force-density option here.

The curved washer sits at the other extreme. It's a single gentle curve rather than a true cone or wave, the simplest and cheapest of the five to produce, and by far the lowest-force option. It exists almost entirely to take up assembly slack: light bearing preload, anti-rattle, a small constant push against a race, nothing that needs real clamping force.

The wave washer (typically three waves around the circumference, single turn) is a step up from the curved washer: more force for the same material, at a moderate deflection, still a fraction of a Belleville's force density. It's the common drop-in for light-duty bearing preload where a Belleville would be overkill.

Nest wave washers crest-to-crest and you get a stacked wave spring: a compact substitute for a coil spring, with more total deflection and a more linear rate than any single washer in this family, in a much smaller radial envelope than a coil spring would need for the same force (though it does need more axial height than a single washer or a short Belleville stack).

The finger spring washer replaces the continuous wave or cone with cantilevered fingers. Force output is comparatively low, and the fingers are shaped to hold that force nearly constant across the working travel, which is exactly what steady electrical contact pressure needs through vibration and thermal cycling. If the joint also has to carry current or hold an EMI/RF shield path, this is usually the only one of the five actually built for that job.

Card 05's guide-pin/bore friction (contact (c) in the friction formulas there) is the one friction source that isn't reliably predictable from geometry — it dominates in long stacks and grows with taper angle, and a poorly guided stack can wear unevenly or walk sideways under load. The alternative is to build the centering into the washers or the stack hardware themselves, so the stack locates itself without sliding against a separate rod or bore at all. Bauer's catalog gives three ways to do this.

Each washer (or its mating end cap) is machined with an integral cylindrical shoulder that nests directly into the adjacent washer, locating the stack radially without any separate guide rod or sleeve. Because the shoulder itself requires machining, this is most practical on washers that are already fully machined on all faces — DIN 2093 Group 3, contact-surface springs (card 07) — where the extra step doesn't add a new manufacturing operation.

Centering Using a Cylindrical Shoulder

Separate rings, typically a shallow T-shaped cross-section, are inserted between alternating washers in the stack. Each ring locates against the O.D. or I.D. contact points of the washers on either side of it, holding the stack in radial alignment without requiring any machining on the washers themselves — a good option when the washers are off-the-shelf, unmachined parts.

Intermediate Centering Rings

A shallow ring-shaped groove is machined near the O.D. (or I.D.) contact face of each washer. A set of small balls, or a segmented wire ring, sits in the groove between adjacent washers and provides the radial location — the balls or wire roll rather than slide, so this is the lowest-friction of the three self-centering methods, at the cost of the extra groove-machining and loose-parts handling.

Ball / Wire-Ring Centering

As a quick allowance, use about 2% to 3% per sliding surface. Use the friction equations above when you need more.

More parallel discs mean more rubbing. Heat from friction can cut fatigue life.

Common guidance is a few discs in parallel, often no more than four. Use a solid lubricant such as molybdenum disulfide.

Keep the stack height at or below three times the OD. Also keep it to about ten discs in series. Short stacks are more efficient.

Life depends on the deflection range. Keep dynamic deflection to about 75% to 80% of the maximum.

A higher final load raises stress and cuts life. A higher preload cuts the swing and adds life.

The h/t ratio shapes the curve and the fatigue life.