Disc Spring Calculator for Force, Characteristic and Stack

Which dimensions govern a disc spring?

A disc spring is a conical annular disc that is flattened under axial load. Outside diameter De and inside diameter Di set the diameter ratio δ = De/Di, the annular width and the guidance; the thickness t enters the force to the third power and the stresses to the second. The cone height h₀ is the height of the unloaded cone excluding thickness, the free height l₀ = h₀ + t the total height of the single disc; the deflection s is the flattening of one disc from its free position, and at the flat position s = h₀. Discs with bearing surfaces (series C and thick discs) are calculated with the reduced thickness t′ and h′₀ = l₀ − t′. Dimensions, materials, test forces and quality requirements of the standard series A, B and C are given in EN 16983; all calculation equations come from EN 16984.

Formula and variables

F = 4E/(1 − μ²) · t⁴/(K1·De²) · K4² · s/t · [K4² · (h₀/t − s/t)·(h₀/t − s/(2t)) + 1]

• δ = De/Di; K1, K2, K3 = f(δ) (EN 16984, 5.3); K4 = 1 without bearing surfaces
• Fc = F(s = h₀) = 4E/(1 − μ²) · t³·h₀/(K1·De²) · K4² (5.4)
• σOM = −4E/(1 − μ²) · t²/(K1·De²) · K4 · s/t · 3/π (5.5)
• R = dF/ds (5.6); W = ∫F ds (5.7)
• Fges = n · F; sges = i · s; L₀ = i · [l₀ + (n − 1)·t] (clause 7)
• Fges,R = F · n / (1 ∓ wM·(n − 1) ∓ wR) (− loading, + unloading; clause 8)

Choose the inputs correctly

Outside diameter De, inside diameter Di, thickness t and free height l₀ (giving h₀ = l₀ − t) describe the single disc; for discs with bearing surfaces also the reduced thickness t′. Total deflection, the number of equally oriented discs n per pack, the number of opposed packs i and the friction coefficients wM (surface friction) and wR (edge friction) describe the column. The cone-height-to-thickness ratio shapes the nonlinear characteristic. Parallel discs raise force and series stacks raise travel; stack friction causes hysteresis so loading and unloading differ.

How to use the calculator

Enter De, Di, t and l₀ of the disc – for standard discs from EN 16983 Table 5 (series A, h₀/t ≈ 0.75) or Table 6 (series B, h₀/t ≈ 1.3) – plus the desired total deflection and the arrangement: n discs equally oriented in parallel per pack, i packs opposed in series. wM and wR give the stack its friction hysteresis; set both to 0 for the frictionless curve. The calculator returns single-disc force, stack force, loading and unloading force, spring rate, spring work, stresses and installed lengths.

Worked example

Disc spring series A 40 × 20.4 × 2.25 (l₀ = 3.15 mm, h₀ = 0.9 mm) at s = 0.675 mm = 0.75·h₀: F = 6,500 N, σOM = −1,196 MPa, σII = 1,328 MPa, Fc = 8,456 N – identical to the test force and stresses in EN 16983 Table 5. Column of i = 3 packs with n = 2 discs each at sges = 2.025 mm: Fges = 13,000 N, L₀ = 16.2 mm, loaded length 14.175 mm; with wM = 0.01 and wR = 0.03 the force is 13,542 N on loading and 12,500 N on unloading.

How to read single-disc and stack results

The single-disc force F belongs to the deflection s of one disc and follows the nonlinear characteristic of EN 16984 clause 5.4; the test force Ft is the value at s = 0.75·h₀. For a pack of n equally oriented discs Fges = n·F at the same deflection, for a column of i opposed packs sges = i·s at the same force. The calculator additionally reports the force on loading and on unloading: per EN 16984 clause 8, friction at the conical surfaces (wM) and at the bearing edges (wR) raises the force on loading and lowers it on unloading – the difference is the hysteresis of the stack. The calculated stresses σOM and σI to σIV are nominal values (EN 16984 clause 9): σOM is used to assess the permissible free height and should not exceed about Rm, while the tensile stresses on the underside (location II inside, III outside) govern fatigue strength.

Diameters, thickness, heights and deflections in millimetres, forces in N, spring rate in N/mm, spring work in N·mm, stresses in MPa; E = 206,000 MPa and μ = 0.3 for spring steel to EN 16983.

Disc spring characteristic, stack and column

The characteristic of a single disc spring is nonlinear per EN 16984 clause 6; its shape depends on the ratio h₀/t. Small values around 0.4 give an almost linear, rising curve, around h₀/t ≈ 1.4 it becomes nearly horizontal in the middle range, and above that it passes through a falling branch. The equation idealises the disc as a conical shell of constant section; residual stresses from manufacturing, dimensional tolerances and the real contact geometry at the edges make the measured curve deviate from it. EN 16983 therefore characterises the discs by the test force at s = 0.75·h₀ with tolerances, and the calculator flags this deflection as the recommended limit.

A pack of n equally oriented discs behaves like a thicker disc: same deflection, n times the force. A column of i opposed packs adds the deflections at the same force. Combined, EN 16984 clause 7 gives Fges = n·F, sges = i·s and the installed length L₀ = i·[l₀ + (n − 1)·t]. For opposed discs with h₀/t above about 1.25 the deflection distributes unevenly between the discs, which can lead to functional failure; discs of different thickness or packs with increasing disc count give a progressive characteristic – see the progressive disc spring stack calculator.

Friction is part of the force calculation per EN 16984 clause 8: it occurs at the conical surfaces of touching discs (wM) and at the flat bearing plates (wR), grows with the number of discs per pack and depends on surface and lubrication. The calculator applies Fges,R = F·n/(1 ∓ wM·(n − 1) ∓ wR) and reports both branches. A fatigue assessment is not part of this characteristic calculation: for dynamically loaded stacks (from 10⁴ cycles per EN 16984 clause 10) the upper and lower stresses at location II or III have to be checked against the fatigue diagrams – use the disc spring fatigue calculator. Other spring types are in the spring cluster.

Typical applications

Designing a disc spring stack for a desired, often strongly nonlinear force-deflection characteristic: preload of fasteners and bearings, compensation of settling and thermal expansion, overload clutches, clamping and tensioning elements, and snap mechanisms with h₀/t in the falling range of the curve.

Assumptions, limits and common mistakes

EN 16984:2016 (DIN EN 16984:2017-09) – calculation: clause 5.1 scope, 5.3 deflection factors, 5.4 spring force, 5.5 calculated stresses, 5.6 spring rate, 5.7 spring work, 7 arrangement, 8 friction, 9 stress assessment, 10 load types. EN 16983:2016 (DIN EN 16983:2017-09) – quality requirements and dimensions: Tables 5 and 6 (series A, B, C with test force and stresses). Roloff/Matek, Maschinenelemente, 27th edition, ch. 10.3.5 disc springs, Fig. 10.16 (characteristic shapes as a function of h₀/t). Limits: 16 < De/t < 40 or 1.8 < De/Di < 2.5 and materials to EN 16983; recommended deflection up to 0.75·h₀; for h₀/t above about 1.25 the deflection of opposed discs distributes unevenly; set, relaxation, temperature and fatigue life are not assessed here.

Common mistake: Do not confuse h₀ and l₀: the tables list the free height l₀, the cone height is h₀ = l₀ − t. Keep parallel and series apart: discs in parallel (equally oriented) multiply the force, discs in series (opposed) multiply the deflection. Do not plan deflections above 0.75·h₀ – beyond that the real force deviates increasingly from the equation and the flat position is approached. And do not forget friction: a pack of many parallel discs can need considerably more force on loading than the ideal characteristic suggests.