How to use this calculator
- Enter the disc dimensions. Outside diameter De, inside diameter Di, thickness t and overall free height l0 from the drawing or catalog row. The cone height h0 = l0 - t is derived.
- Set the working deflection. Deflection s is per disc. Use 0.75 × h0 to compare directly with DIN 2093 catalog load values.
- Describe the stack. Discs nested in parallel multiply load; alternating packets in series multiply travel. Leave both at 1 for a single washer.
- Check material constants. E = 206,000 N/mm² and μ = 0.3 match DIN 2092 spring steel. Enter supplier values for stainless, Inconel or other alloys.
- Read load, stress and shape checks. Use load at deflection and at flat, the OM set-stress band, h0/t characteristic shape and the stack outputs to pick or verify a disc.
How it works
A Belleville washer (conical disc spring) carries load by flattening its
cone. The standardized calculation comes from Almen and Laszlo (1936) and
is the method behind DIN 2092. Everything is driven by two shape numbers:
the diameter ratio δ = De/Di, which sets the coefficients
K1-K3, and the characteristic ratio h0/t, which sets the
shape of the load-deflection curve:
F(s) = (4E/(1-μ²)) · (t⁴/(K1·De²)) · (s/t) · [(h0/t - s/t)(h0/t - s/2t) + 1]
With h0/t around 0.4 the curve is nearly linear; by 1.41 it
develops a flat plateau, and beyond that it is regressive — the load
falls with extra travel, which is also how snap-action discs work. The
calculator reports the local rate dF/ds at your working
point, the load at flat, and the OM stress that DIN 2093 uses to keep
permanent set in check.
Stacking extends a single disc: nested parallel discs multiply force, alternating series packets multiply travel. For the surrounding joint or preload problem, pair this with the bolt preload & torque calculator or check spring alternatives with the compression spring calculator.
Worked example
Verified against the live calculator
A disc with De = 40 mm, Di = 20.4 mm,
t = 2.25 mm and l0 = 3.15 mm (DIN 2093 A 40
proportions) has cone height h0 = 0.9 mm and
h0/t = 0.4. At the DIN reference deflection
s = 0.75 × h0 = 0.675 mm with spring steel
(E = 206,000 N/mm², μ = 0.3), the load is
F ≈ 6,500 N, the load at flat is ≈ 8,456 N,
the local rate is ≈ 8,784 N/mm, and the stresses are
σOM ≈ 1,196 N/mm² (inside the set-resistant band) and
σI ≈ 2,086 N/mm² compressive. Two discs in parallel and six
packets in series would carry ≈ 13,000 N over
4.05 mm of travel with a stack free length of
32.4 mm.
Spring material data
The DIN 2092 coefficients and formulas implemented by this calculator:
| Quantity | Formula | Notes |
|---|---|---|
| Diameter ratio | δ = De / Di | Common disc springs run δ ≈ 1.7-2.6; DIN series sit near 2. |
| K1 | K1 = (1/π)·((δ-1)/δ)² / ((δ+1)/(δ-1) - 2/ln δ) | Load-formula geometry coefficient (≈ 0.69 at δ = 2). |
| K2 | K2 = (6/π)·((δ-1)/ln δ - 1) / ln δ | Edge-stress coefficient (≈ 1.22 at δ = 2). |
| K3 | K3 = (3/π)·(δ-1) / ln δ | Edge-stress coefficient (≈ 1.38 at δ = 2). |
| Load at deflection | F(s) = (4E/(1-μ²))·(t⁴/(K1·De²))·(s/t)·[(h0/t - s/t)(h0/t - s/2t) + 1] | Single disc without contact flats, t′ = t. |
| Load at flat | F(h0) = (4E/(1-μ²))·(t³·h0/(K1·De²)) | The bracket collapses to 1 at s = h0. |
| Stress at OM | σOM = (4E/(1-μ²))·(t²/(K1·De²))·(s/t)·(3/π) | DIN 2093 permanent-set check; keep within the guidance band. |
| Edge stress I | σI = (4E/(1-μ²))·(t²/(K1·De²))·(s/t)·[K2·(h0/t - s/2t) + K3] | Compressive stress at the upper inner edge. |
| Stack | F = n·F₁, s = i·s₁, L0 = i·(l0 + (n-1)·t) | Parallel multiplies load, series multiplies travel; friction neglected. |
Source: J.O. Almen & A. Laszlo, 'The Uniform-Section Disk Spring', ASME 58-10 (1936); DIN 2092 (calculation) and DIN 2093 (dimensions, quality groups). Verify catalog loads and set limits with the disc-spring supplier.
Frequently asked questions
How do I calculate Belleville washer load?
Use the DIN 2092 (Almen-Laszlo) formula: F(s) = (4E/(1-μ²)) · (t⁴/(K1·De²)) · (s/t) · [(h0/t - s/t)(h0/t - s/2t) + 1], where t is thickness, h0 = l0 - t the cone height, De the outside diameter and K1 a coefficient of the diameter ratio De/Di. This calculator evaluates it directly.
Why is a disc spring rate non-linear?
The load curve depends on the cone geometry while it flattens. The shape is set by h0/t: below about 0.4 the characteristic is nearly linear, around 1.41 it has a flat plateau, and above 1.41 it becomes regressive and can snap through. Quote disc springs by load at deflection, not by one rate.
What is the difference between series and parallel stacking?
Parallel (nested, same direction) discs multiply force: F = n·F_single. Series (alternating) packets multiply travel: s = i·s_single. Both together give a stack free length L0 = i·(l0 + (n-1)·t). Friction is neglected here but grows with each parallel interface.
What deflection should a Belleville washer run at?
DIN 2093 tabulates characteristics at s = 0.75·h0 and most catalogs recommend staying at or below ~75% utilization for static service so the load tolerance and set stay controlled. The calculator flags working points beyond 75% and at flat.
Does this cover discs with contact flats?
No. DIN 2093 series 3 discs (t > 6 mm) have machined bearing flats and use a reduced thickness t′ in a modified formula. This page models standard discs without contact flats, t′ = t.
Can I check fatigue with this calculator?
Only partially. It reports the compressive stresses at OM and at the upper inner edge (point I) used for static and set checks. Fatigue life is governed by the tensile stresses at points II and III against supplier S-N diagrams, which stay with the disc-spring maker.
Method & assumptions
- Almen-Laszlo / DIN 2092 uniform-section model: linear-elastic material, uniform thickness, load applied at the edges through the full circumference, t′ = t (no machined contact flats).
- DIN 2093 series 3 discs (t > 6 mm) use a reduced thickness t′ and shifted bearing edges; their catalog loads will not match this page.
- Stack outputs neglect friction. Real parallel packets gain a few percent load per interface on loading and lose it on unloading (hysteresis); series stacks need a guide rod, and long stacks shift load share.
- σOM and σI are reported as magnitudes and are compressive. The tensile fatigue stresses at points II/III, supplier S-N limits, edge condition and preset are not modeled.
- Static verdict bands (75% utilization, σOM ≈ 1200/1600 N/mm²) are customary DIN 2093-based guidance for spring steel, not a substitute for supplier data.