Disc Spring Calculator (Belleville)
Calculate the force, characteristic curve and factors of a single Belleville disc spring to DIN 2093 using the Almen-Laszlo approach. Enter geometry (outer and inner diameter, thickness, cone height), material and deflection – the calculator returns the nonlinear force-deflection curve, factors K1 to K3, the edge stress as an approximation and the check against the flattening limit with a traffic-light rating, live with every input.
Disc Spring Calculator (DIN 2093)
Model: single disc spring to DIN 2093, Almen-Laszlo approach, linear-elastic, without contact flats (factor K4 = 1). The edge stress σ_OM is an approximation; a fatigue and set check is not included. For spring columns multiply force (same direction) or deflection (alternating) by the count.
Results
Calculating …
Formulas and fundamentals
The basis is the Almen-Laszlo approach underlying DIN 2093. From the diameter ratio delta = De/Di follow the dimensionless factors K1 = (1/π)·((delta-1)/delta)²/((delta+1)/(delta-1) - 2/ln delta) as well as K2 and K3, which depend on delta via ln(delta). K1 governs the spring force, K2 and K3 the edge stresses. For the common geometry delta ≈ 2, K1 is close to 0.69.
The spring force depends nonlinearly on deflection: F(s) = 4·E/(1-nu²)·t⁴/(K1·De²)·(s/t)·[(h0/t - s/t)·(h0/t - s/(2t)) + 1]. The bracketed term produces the characteristic degressive-progressive shape. When flattened at s = h0 the bracket becomes exactly 1, so F(h0) = 4·E/(1-nu²)·t⁴/(K1·De²). The height ratio h0/t determines the shape: nearly linear at small values, with a descending branch (force peak before flat) for h0/t > √2.
The edge stress at the most highly stressed point OM (top side at the inner edge) is approximated by sigma_OM = -4·E/(1-nu²)·t²/(K1·De²)·(s/t)·[K2·(h0/t - s/(2t)) + K3]; it is a compressive stress and rises toward the flat position. The usable deflection ends at the flattening limit s = h0; working points up to about s ≈ 0.75·h0 are common to limit set and stress peaks.
Worked example
A disc spring with De = 50 mm, Di = 25.4 mm, thickness t = 2 mm and cone height h0 = 1.3 mm (overall height l0 = 3.3 mm) made of spring steel (E = 206,000 N/mm², nu = 0.3). The diameter ratio is delta = 50/25.4 = 1.969, giving K1 = 0.688 and the height ratio h0/t = 0.65.
When flattened at s = h0 = 1.3 mm the bracket equals 1 and the spring force is F(h0) = 4·206000/(1-0.3²)·2⁴/(0.688·50²) = 5477 N. At s = 0, F = 0; in between the curve is nonlinear, with a decreasing secant spring rate toward the flat point.
For a working point at s = 0.75·h0 = 0.975 mm the force is about 4380 N. The deflection utilization s/h0 = 75 percent is in the green range of the traffic light. The example shows the typical advantage of a disc spring: high forces in a small space with a tunable, nonlinear characteristic set by the height ratio h0/t.
Frequently asked questions
Why is a disc spring characteristic nonlinear?
Because the cone angle changes as the spring is compressed and it becomes progressively flatter. The bracketed term in the Almen-Laszlo formula captures this. The height ratio h0/t sets the shape: nearly linear for small h0/t, and for h0/t > √2 it has a descending branch with a force peak before the flat position.
What does the flattening limit s = h0 mean?
At s = h0 the spring is fully flattened and the cone has gone to zero. The bracketed term is then 1 and the force reaches F(h0). Compressing beyond h0 is only possible with a steep rise in stress and is avoided; typical working points are around 0.75·h0.
How can I tune force and deflection?
Disc springs are stacked into columns. Springs stacked in the same direction (parallel) multiply the force, springs stacked in alternating directions (in series) multiply the deflection. The calculator handles the single spring; a column follows by multiplying force or deflection by the respective count.
Which material properties apply?
For spring steel to DIN 2093, E = 206,000 N/mm² and Poisson's ratio nu = 0.3 are used; both are preset here and can be overridden. The elastic modulus enters the force linearly, Poisson's ratio via the factor 1/(1-nu²).
Does the calculator replace a strength and fatigue check?
No. The calculator provides force, characteristic curve and an approximation of the edge stress sigma_OM. A fatigue check across the stress points, the effect of contact flats (factor K4 for thick series-C springs) and set behaviour are not included. For a service-strength check refer to the Goodman diagrams of DIN 2093.
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