Belleville Disc Spring Calculator: DIN 2092/2093 Force & Stack
Under DIN EN 16984 (DIN 2092), a Belleville disc spring's load is governed by the non-linear Almen–Laszlo equation; nesting np discs in parallel multiplies force (np·F), while stacking is packets in alternating series multiplies travel (is·s). A 2-parallel × 12-series stack of DIN 2093 A 50 × 25.4 × 2.5 mm discs at 0.75h0 delivers 13.99 kN (3,144 lbf) at 9.90 mm travel.
DIN 2092 / 2093 Belleville Disc Spring & Parallel/Series Stack Calculator
Compute non-linear single-disc load F(s) by the DIN 2092 Almen–Laszlo equation, geometry factor K1, cone-to-thickness ratio h0/t, and total stack force, stroke, and loaded length for np parallel discs across is series packets.
How Does the DIN 2092 (Almen–Laszlo) Equation Calculate Disc Spring Force?
Conical disc springs (standardized in DIN EN 16983 / DIN 2093 and analyzed in DIN EN 16984 / DIN 2092) deliver massive axial preload inside tight radial envelopes—such as CNC machining-center HSK/CAT automatic tool-changer (ATC) drawbars (ASME B5.50 (V-Flange Tool Shanks)), fail-safe caliper brakes, and live-loaded high-temperature flange bolts (NASA-RP-1228 (Fastener Design Manual)). Given outer diameter De, inner bore diameter Di, thickness t, and free cone height h0 = l0 − t in 51CrV4 spring steel (E = 206,000 MPa, ν = 0.30, tested per ISO 6892-1 (Metallic Tensile Testing)), the diameter ratio is δ = De / Di. The Almen–Laszlo constant K1 = (1/π)·[((δ−1)/δ)² / ((δ+1)/(δ−1) − 2/ln(δ))] combines with the cubic polynomial (s/t)[(h0/t − s/t)(h0/t − s/2t) + 1] to give the exact load at any deflection s, matching engineering handbooks from Schnorr GmbH, Christian Bauer, and Machinery's Handbook (31st Ed., Industrial Press).
The dimensionless ratio h0 / t controls the shape of the force-deflection curve: DIN Series A heavy-duty springs (h0/t ≈ 0.44, such as our 50 × 25.4 × 2.5 mm disc where h0/t = 0.440) behave almost linearly, whereas Series B discs (such as 50 × 25.4 × 2.0 mm with h0 = 1.40 mm, h0/t = 0.700, and F1 = 4,762 N) and Series C discs (h0/t ≈ 1.30) exhibit a strongly digressive curve that approaches a constant-load horizontal plateau near flat position (h0/t → √2 = 1.414). Force and length conversions follow NIST SI Units Reference and NIST Special Publication 811.
| DIN 2093 Designation (De×Di×t) | Thickness t / Cone h0 | Factor K1 (δ=De/Di) | Deflection s0.75 | Force at s = 0.75·h0 | Flat Load Fc (s=h0) |
|---|---|---|---|---|---|
| DIN 2093 — A 20 × 10.2 × 1.1 | 1.1 / 0.45 mm (h0/t=0.41) | 0.6861 (δ=1.96) | 0.338 mm | 1,521 N (342 lbf) | 1,976 N |
| DIN 2093 — A 28 × 14.2 × 1.5 | 1.5 / 0.65 mm (h0/t=0.43) | 0.6885 (δ=1.97) | 0.488 mm | 2,841 N (639 lbf) | 3,680 N |
| DIN 2093 — A 35.5 × 18.3 × 2.0 | 2.0 / 0.80 mm (h0/t=0.40) | 0.6815 (δ=1.94) | 0.600 mm | 5,187 N (1,166 lbf) | 6,747 N |
| DIN 2093 — A 50 × 25.4 × 2.5 | 2.5 / 1.10 mm (h0/t=0.44) | 0.6878 (δ=1.97) | 0.825 mm | 6,994 N (1,572 lbf) | 9,051 N |
| DIN 2093 — B 50 × 25.4 × 2.0 | 2.0 / 1.40 mm (h0/t=0.70) | 0.6878 (δ=1.97) | 1.050 mm | 4,762 N (1,071 lbf) | 5,898 N |
| DIN 2093 — A 63 × 31.0 × 3.5 | 3.5 / 1.40 mm (h0/t=0.40) | 0.7006 (δ=2.03) | 1.050 mm | 15,025 N (3,378 lbf) | 19,545 N |
| DIN 2093 — A 80 × 41.0 × 4.0 | 4.0 / 1.70 mm (h0/t=0.42) | 0.6841 (δ=1.95) | 1.275 mm | 17,354 N (3,901 lbf) | 22,503 N |
| DIN 2093 — A 100 × 51.0 × 6.0 | 6.0 / 2.20 mm (h0/t=0.37) | 0.6861 (δ=1.96) | 1.650 mm | 48,022 N (10,796 lbf) | 62,711 N |
How Do Parallel and Series Stacks Multiply Force, Stroke, and Stack Length?
To customize both clamping force and axial knockout stroke using a single standard DIN 2093 washer size, engineers combine np discs nested face-to-back in parallel within each packet and stack is packets back-to-back in alternating series. Total stack force is Fstack = np × F1, total stack deflection is sstack = is × s1, and total unloaded stack length is L0,stack = is × (np·t + h0). Always restrict continuous working deflection to s ≤ 0.75·h0 (because contact-edge rolling beyond 75% causes actual force to spike above the theoretical curve and induces fatigue cracking) and never nest more than np = 3 discs in parallel to avoid excessive sliding friction hysteresis.
| Stack Arrangement (np × is) | Force Scaling | Travel Scaling | Stack Force at 0.75·h0 | Stack Travel stot | Free L0 → Loaded L1 |
|---|---|---|---|---|---|
| Single disc (1 parallel × 1 series) | 1× F1 | 1× s1 | 6.99 kN (1,572 lbf) | 0.82 mm | 3.60 → 2.77 mm |
| 2 in parallel — nested (( )) | 2× F1 (+~3% friction) | 1× s1 | 13.99 kN (3,144 lbf) | 0.82 mm | 6.10 → 5.27 mm |
| 3 in parallel — nested ((( ))) | 3× F1 (+~6% friction) | 1× s1 | 20.98 kN (4,717 lbf) | 0.82 mm | 8.60 → 7.77 mm |
| 4 in series — opposed ()()() | 1× F1 | 4× s1 | 6.99 kN (1,572 lbf) | 3.30 mm | 14.40 → 11.10 mm |
| Spindle drawbar pack (2 parallel × 12 series) | 2× F1 | 12× s1 | 13.99 kN (3,144 lbf) | 9.90 mm | 73.20 → 63.30 mm |
| Heavy clamp stack (3 parallel × 8 series) | 3× F1 | 8× s1 | 20.98 kN (4,717 lbf) | 6.60 mm | 68.80 → 62.20 mm |
Worked Example: Sizing a 24-Disc CNC Spindle Drawbar Stack
A vertical machining center spindle drawbar uses 24 discs of DIN 2093 A 50 × 25.4 × 2.5 mm (h0 = 1.10 mm, K1 = 0.6878) arranged in is = 12 alternating series packets of np = 2 parallel discs compressed to s / h0 = 0.75 (s1 = 0.825 mm). A single disc produces 6,994 N (flat load Fc = 9,051 N), so the 2-parallel packet doubles drawbar retention force to 13,987 N (13.99 kN / 3,144 lbf). Across 12 series packets, unloaded stack length is 73.20 mm, total stroke is 9.90 mm, and clamped length is 63.30 mm. Compare spindle retention interfaces on our CAT40 / BT40 / HSK63A spindle taper chart, check high-speed rotor loads on the CNC toolholder balance G2.5 calculator, verify live-loaded flange preload on the master bolt torque chart, and size guide-pin fits on the ISO 286 hole and shaft tolerance calculator.
Cite this page
ToolSpecVault Engineering Data Team. (2026, October 29). Belleville Disc Spring Calculator: DIN 2092/2093 Force & Stack. ToolSpecVault. https://toolspecvault.com/belleville-disc-spring-stack-force-deflection-calculator/