Design a coned disc spring (Belleville washer). Adjust the outside and inside diameter, thickness, free height and deflection to see the spring force, spring rate and load-deflection characteristic update in real time, and explore the nonlinear S-shaped curve set by the height ratio h0/t.
Parameters
Outside diameter De
mm
Inside diameter Di
mm
The ratio delta = De/Di sets the coefficient K1
Plate thickness t
mm
Free cone height h0
mm
Cone height at no load. h0/t sets the nonlinearity
Young's modulus E
GPa
About 206 GPa for spring steel. Poisson's ratio v = 0.3 fixed
Deflection s
mm
Compression travel. At s = h0 the spring is fully flat
Results
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Spring force F (N)
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Flat load F_flat (N)
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Spring rate k (N/mm)
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Diameter ratio δ
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Height ratio h0/t
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Characteristic type
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Disc-spring cross-section — compression animation
The coned disc spring is compressed between two flat platens, its cone angle reducing as it flattens. Colour shows the characteristic type (green = near-linear / orange = nonlinear / red = unstable region).
Spring force F [N] from the Almen-Laszlo formula. E: Young's modulus, ν: Poisson's ratio (0.3), t: thickness, De: outside diameter, h0: free height, s: deflection. The bracketed term is quadratic in s, so the characteristic is nonlinear.
Spring rate k is the slope of load against deflection; the flat load F_flat is the force when the disc is pressed completely flat. The height ratio h0/t sets how nonlinear the characteristic is.
What is a Belleville Spring?
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A "Belleville spring" is that metal washer with a hole in the middle that's slightly dished, like a little plate, right? Is that really a spring?
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Yes — also called a disc spring. It looks like a washer, but it is a genuine spring. A flat plate is formed into a shallow cone (a plate shape), and when you press it from the top the cone collapses towards flat, and that restoring force is the spring force. Unlike a coil spring, it is thin and compact yet can deliver a very large load.
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I see. But when I move "deflection s" on the left, the force F doesn't increase at a steady rate. Sometimes it barely grows at all... is that a bug?
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No, that is the most interesting thing about a disc spring — not a bug. A coil spring follows "load = spring rate × deflection" in a clean proportion, but a disc spring changes its very shape as the cone collapses, so load and deflection are not proportional. Look at the "load-deflection characteristic" chart below — it is curved, not straight. The shape of that curve is set by the "height ratio h0/t", the free height divided by the thickness.
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h0/t, then. How does that curve change as I increase the height ratio?
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When h0/t is small (below 0.4) the curve is almost straight, close to a coil spring. Around 0.4-1.3 it bends gently — a "progressively nonlinear" curve. Above 1.3 the curve lies down and you get an "S-shape" where the load barely rises over a stretch. Above 2.83 you even get an "unstable region" where the load actually drops as you push deeper — a negative spring rate. Raise h0 on the left and watch the characteristic-type card change colour.
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A spring whose load drops as you push harder sounds wrong. What is that good for?
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That is actually an important use. By working in the flat region where the load is "constant", you can hold the clamping force almost steady even as the mating part wears or expands and contracts with heat. Bolt loosening prevention and clamps on large machines are good examples. Because the force barely changes even when deflection shifts a little, you can stabilize the preload on a gasket or bearing. "Curved" is not a defect — the designer is deliberately exploiting that curvature.
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If one disc can vary this much, stacking several should let you do even more, right?
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That is exactly the strength of disc springs. Stacked in the same direction (parallel), the load multiplies by the count and it gets stiff. Stacked alternately (series), the deflection multiplies and it gets soft. Combine them and you can reach the load and stroke you want even in a tight mounting space. This tool computes one disc, so when stacking, convert load and deflection by the number of discs.
Frequently Asked Questions
The force of a Belleville (disc) spring is given by the Almen-Laszlo formula F = (4E/(1-v²))·(t⁴/(K1·De²))·(s/t)·[(h0/t-s/t)(h0/t-s/(2t))+1]. Here E is Young's modulus, v is Poisson's ratio (0.3), t is the plate thickness, De is the outside diameter, h0 is the free cone height and s is the deflection. K1 is a dimensionless coefficient that depends only on the diameter ratio delta = De/Di. This tool evaluates this formula for any deflection s.
A disc spring is a plate formed into a shallow cone, and as it deflects the cone angle changes so the cross-sectional geometry changes continuously. Unlike a coil spring, load and deflection are not proportional. The bracketed term of the Almen-Laszlo formula is quadratic in s, so the load-deflection curve is curved. Its shape is governed by the height ratio h0/t: a low h0/t gives a nearly linear curve, while a high h0/t produces an S-shaped curve where the load stays constant or even decreases.
The larger the height ratio h0/t (free height divided by thickness), the stronger the nonlinearity. For h0/t < 0.4 the curve is nearly linear; 0.4-1.3 gives a progressively nonlinear curve; 1.3-2.83 produces a strong S-shape with a near-constant-load plateau; and h0/t >= 2.83 introduces an unstable region where the load actually drops as deflection increases (negative spring rate). The constant-load region is deliberately used to keep clamping force steady despite part wear or thermal expansion.
Disc springs let you tune the characteristic by stacking. Stacked in the same direction (in parallel) the load multiplies by the number of discs while deflection stays the same (stiffer). Stacked alternately (in series) the load stays as for one disc while deflection multiplies by the count (softer). Combining parallel and series stacks lets you reach the required load and deflection (stroke) within a limited mounting space. This tool models a single disc, so for a stack convert load and deflection by the number of discs.
Real-World Applications
Bolt-joint loosening prevention and preload retention: A disc spring under the seat of a large bolt absorbs the slight stretch and contraction of the bolt due to thermal expansion or creep, holding the clamping force almost constant. Working in the flat region of the S-shaped curve means the load barely changes even as deflection shifts, giving a joint that resists loosening even under vibration. Belleville springs are widely used on pipe-flange and pressure-vessel bolts.
Clutch and brake clamping mechanisms: An automotive clutch cover uses a large-diameter disc spring (the diaphragm spring) to press the clutch plate with a steady force. If the operating point sits on the flat region, the clamping force does not fall much even as the friction material wears, keeping the transmitted torque stable. Here the nonlinear characteristic of the disc spring is the very key to product performance.
Bearing preload and clearance adjustment: Disc springs are built in to apply a constant preload to rolling bearings. When the shaft elongates with temperature, the disc spring absorbs the change and keeps the internal bearing clearance correct. Because it delivers a large load in a thinner, more compact package than a coil spring, it fits easily into limited axial space — a common sight on machine-tool and motor spindles.
Overload protection and energy absorption: In press dies and safety devices, disc springs are stacked to obtain a large load and stroke and serve as a cushioning or relief mechanism when an excessive force is applied. A high-h0/t disc spring with a negative spring rate snaps through suddenly once a certain load is exceeded, so it is also applied to the design of toggle mechanisms and the tactile detent of switches.
Common Misconceptions and Pitfalls
The biggest misconception is "a disc spring follows a proportional load-deflection law like a coil spring". The load-deflection curve of a disc spring is intrinsically nonlinear, and the spring rate k varies with deflection s. If you measure k at a single deflection and treat it as "the spring rate of this disc spring", the load will be far off at a different operating point. When designing a disc spring, always look at the whole characteristic curve over the working range and be aware of whether you are using the flat region or the rising region. At high h0/t there is even a region where k is negative.
Next, "trusting the Almen-Laszlo load directly". The formula used here is an approximation that assumes an ideal disc spring with a small cone angle and uniform thickness. A real disc spring has residual stress from forming, edge chamfers (discs with flats), hysteresis from friction, and stress under cyclic load (concentrated on the upper inner edge and the lower outer edge), so the calculated and measured loads differ. For critical applications, always check against the manufacturer's measured catalogue values, the DIN 2092/2093 standards, or finite-element analysis.
Finally, the hasty assumption that "pressing all the way to s = h0 gives the maximum load". At full flattening (s = h0) the bracketed term becomes one and you get the flat load F_flat, but the stress on the upper inner edge is then at its peak and becomes the origin of fatigue failure under repeated use. In practice, take about 75-85% of the flat-load deflection as the upper working limit, and avoid "over-flattening" past s > h0. This tool shows a warning when s > h0. With a disc spring it is important to fix the operating point considering both load and stress/life.
How to Use
Enter outer diameter (OD) in mm using deNum slider (typical range 20–80 mm for industrial applications)
Set inner diameter (ID) via diNum slider, ensuring ID < OD to create the annular disc geometry
Adjust thickness (t) in mm and free height (h0) in mm to control cone angle and load characteristics
Review calculated spring force F (N), spring rate k (N/mm), and diameter ratio δ = OD/ID to verify design compliance
Check height ratio h0/t and characteristic type (soft/progressive/hard) to confirm load-deflection behavior
Worked Example
Design a Belleville spring for a bolted joint preload application: OD = 50 mm, ID = 28 mm, thickness = 2.5 mm, free height h0 = 3.2 mm. Diameter ratio δ = 50/28 = 1.79; height ratio h0/t = 3.2/2.5 = 1.28. Using DIN 2093 theory, spring rate k ≈ 18 N/mm, flat load F_flat ≈ 650 N. Under 1 mm deflection, spring force F ≈ 18 N, entering the progressive region typical for fastener locking.
Practical Notes
Stack multiple Belleville springs in series or parallel to achieve desired spring rates and preload forces in bearing preload and vibration damping applications
Height ratio h0/t between 0.8–1.2 produces hard characteristics (steep load curve); ratios >1.5 yield soft springs with shallow load curves
Material selection affects flat load significantly: stainless steel 1.4310 reduces F_flat by 15–20% compared to spring steel C67S, critical for corrosive environments
Verify stress concentration at ID using von Mises stress; typical industrial limit is 1200 MPa for spring steel under full collapse