Roymech engineering encyclopedia

Disc Spring Design


Disc springs, also known as Belleville springs or Belleville washers, are conical annular springs loaded axially. They can produce relatively high forces within a small axial space and may be used individually or combined into stacks to obtain a wide range of spring characteristics.

This page provides engineering reference information for disc spring geometry, force, deflection, stress, correction factors and stacking arrangements. It also covers disc springs manufactured with contact flats.

Not sure whether a disc spring is suitable for your application?
See the Spring Selection and Design Guide for guidance on selecting between disc, compression, extension, torsion, leaf and other spring types.

Contents

Disc Spring Principles

A disc spring is a conical shell spring that deflects when an axial force is applied. Its load-deflection characteristic is determined principally by its outside and inside diameters, thickness, cone height and material properties.

A major advantage of disc springs is that their operating characteristics can be altered substantially by combining otherwise identical discs in different stacking arrangements. This allows the designer to obtain high forces, large deflections or combinations of both without changing the basic disc geometry.

Disc springs arranged individually and in parallel, series and combined stacks

Disc Spring Stacks

Disc springs may be assembled in parallel, series, or in combined series-parallel arrangements. The following ideal relationships neglect friction and other stack effects.

Parallel Stacking

Discs nested in the same direction operate in parallel. If np identical discs are arranged in parallel, each disc undergoes approximately the same deflection and the forces carried by the individual discs are additive.

Fstack = np Fsingle
sstack = ssingle

For an approximately linear portion of the characteristic, the effective spring rate is:

kstack = np ksingle

Parallel stacking is therefore used primarily when a greater force is required for a given deflection.

Series Stacking

Discs facing alternately operate in series. If ns identical discs are arranged in series, each disc carries approximately the same force and the individual disc deflections are additive.

Fstack = Fsingle
sstack = ns ssingle

For an approximately linear portion of the characteristic:

kstack = ksingle ns

Series stacking is therefore used primarily to obtain greater deflection for a given force.

Combined Series and Parallel Stacks

A stack may contain groups of discs in parallel, with those groups arranged in series. For identical discs and equal parallel groups:

Fstack = np Fsingle
sstack = ns ssingle

and, over an approximately linear region:

kstack = np ns ksingle

This gives the designer considerable freedom to adjust force and movement independently while retaining the same basic disc size.

Stack friction: Real disc-spring stacks do not behave as perfectly frictionless springs. Contact between adjacent discs and between the stack and its guides introduces friction and hysteresis. The effect becomes increasingly important with larger stacks and where the load changes repeatedly.

Disc Spring Classification

RoyMech's original reference information used the traditional DIN disc-spring classification according to material thickness:

Group Nominal Thickness Traditional Manufacturing Description
1 t < 1.25 mm Cold formed
2 1.25 mm ≤ t < 6 mm Cold formed; internal and external diameters machined as required and edges rounded.
3 6 mm ≤ t < 16 mm Hot formed; surfaces machined, edges rounded and contact flats provided.

This grouping is retained because it is relevant to the equations and historical RoyMech data below. Current dimensional, quality and calculation requirements should be checked against the applicable current disc-spring standard.

Disc Spring Geometry

Disc spring cross-section and principal dimensions

Disc spring geometry showing outside diameter, inside diameter, thickness and cone height

Disc Spring Nomenclature

Symbol Description
F Axial spring force (N)
s Axial spring deflection (mm)
h Free cone height excluding disc thickness (mm)
h' Adjusted free cone height for a disc with contact flats = H - t' (mm)
H Overall unloaded disc height (mm)
D Outside diameter (mm)
d Inside diameter (mm)
t Nominal disc thickness (mm)
t' Reduced or adjusted thickness used for discs with contact flats (mm)
ν Poisson's ratio (dimensionless)
E Young's modulus (N/mm2)
K1 Geometry factor
K2 Geometry factor
K3 Geometry factor
K4 Correction factor associated with contact flats and reduced thickness
δ Outside-to-inside diameter ratio = D/d
Notation: Poisson's ratio ν is dimensionless. The original RoyMech page incorrectly gave Poisson's ratio units of millimetres.

Disc Spring Geometry Factors

The disc-spring force and stress equations use geometry factors based primarily on the outside-to-inside diameter ratio:

δ = D d

The geometry factors K1, K2 and K3 are calculated as follows:

K1 = ( δ − 1 ) 2 π ( δ + 1 δ − 1 − 2 ln ( δ ) )
K2 = 6 [ δ − 1 ln ( δ ) − 1 ] π ln ( δ )
K3 = 3 ( δ − 1 ) π ln ( δ )

where ln denotes the natural logarithm.

Disc Spring Factor Table

The following factor values are retained from the original RoyMech reference page. The equations above should be used where calculated values are required.

D/d 1.5 1.6 1.7 1.8 1.9 2.0 2.2 2.4 2.6 2.8 3.0 3.5 3.6 3.8 4.0
K1 0.5248 0.5735 0.6131 0.6455 0.6722 0.6943 0.7281 0.7518 0.7684 0.7800 0.7880 0.7979 0.7987 0.7995 0.7994
K2 1.0982 1.1239 1.1488 1.1731 1.1967 1.2198 1.2643 1.3070 1.3482 1.3879 1.4263 1.5178 1.5354 1.5699 1.6037
K3 1.1776 1.1776 1.1776 1.1776 1.1776 1.1776 1.1776 1.1776 1.1776 1.1776 1.1776 1.1776 1.1776 1.1776 1.1776
Legacy table: The numerical table above is retained from the original RoyMech page. The K3 row in particular requires verification because its unchanging value across the complete D/d range is not consistent with the K3 equation above. Until checked, use the equation rather than the tabulated K3 values.

Disc Springs with Contact Flats

Some thicker disc springs are manufactured with annular contact flats at the inside and outside bearing edges. These flats provide improved bearing surfaces and affect the effective geometry and force-deflection characteristic of the disc.

Disc spring cross-section showing annular contact flats

The traditional calculation method compensates for the effect of these contact flats by using a reduced effective thickness t' and an adjusted cone height:

h′ = H − t′

The existing RoyMech information indicates a typical t'/t ratio of approximately 0.94 to 0.96. The exact value should be taken from the applicable specification or spring manufacturer rather than assumed for final design.

A correction factor K4 is used to account for the changed operating characteristic.

K 4 = C 1 2 + ( C 1 2 ) 2 + C 2
C 1 = ( t′ t ) 2 ( 1 4 ⁢ H t + t′ t + 3 4 ) ( 5 8 ⁢ H t + t′ t + 3 8 )
C 2 = C 1 ( t′ t ) 3 ( 5 32 ( H t − 1 ) 2 + 1 )

For a disc spring without contact flats:

K4 = 1

Disc Spring Force

The force produced by a disc spring is non-linear with deflection and depends on the disc geometry, material properties and the ratio of cone height to thickness.

Disc Springs without Contact Flats

For Group 1 and appropriate Group 2 springs without contact flats, K4 = 1.

F = − 4 E s t [ ( h − s 2 ) ( h − s ) + t 2 ] ( 1 − μ 2 ) K 1 D 2

Disc Springs with Contact Flats

For discs with contact flats, the calculation uses the adjusted thickness t', adjusted cone height h' and correction factor K4.

F = − 4 E s K 4 2 [ K 4 2 ( h′ − s 2 ) ( h′ − s ) t′ + t′ 3 ] ( 1 − μ 2 ) K 1 D 2

Disc Spring Stress

A loaded disc spring develops a complex bending-stress distribution. Critical stresses are conventionally evaluated at defined positions around the inner and outer edges of the disc.

The original RoyMech diagram identifies four critical positions. Negative values represent compressive stress and positive values tensile stress.

Stress at position (1)

σ1 = − 4 E K4 s [ K4 K2 ( h − s 2 ) + K3 t ] ( 1 − μ2 ) K1 D2

Stress at position (2)

σ2 = − 4 E K4 s [ K4 K2 ( h − s 2 ) − K3 t ] ( 1 − μ2 ) K1 D2

Stress at position (3)

σ3 = − 4 E K4 s [ K4 ( K2 − 2 K3 ) ( h − s 2 ) − K3 t ] ( 1 − μ2 ) K1 D2 δ

Stress at position (4)

σ4 = − 4 E K4 s [ K4 ( K2 − 2 K3 ) ( h − s 2 ) + K3 t ] ( 1 − μ2 ) K1 D2 δ

These stresses are important both for static loading and when evaluating fatigue under repeated deflection. For cyclic applications also see Fatigue in Springs.

Operating deflection: Do not assume that complete flattening is the normal working condition. Required working deflection, force, stress, fatigue life and relaxation should be considered together. The allowable operating range depends on the spring geometry, material and duty.

Disc Spring Standards

RoyMech's original page referred to DIN 2092 for calculation and DIN 2093 for dimensions and quality. Those references remain relevant to legacy drawings and calculations but have been superseded for current European-standard work.

  • DIN EN 16984:2017-09 – Disc springs – Calculation. This covers calculation and design criteria for single disc springs and disc-spring stacks.
  • DIN EN 16983:2017-09 – Disc springs – Quality specifications – Dimensions. This covers dimensional and quality requirements for disc springs.

The older DIN 2092 calculation standard and DIN 2093 quality/dimensional standard may still be encountered on existing engineering drawings, specifications and components.

Engineering reference: The equations and legacy numerical data on this page are provided for engineering reference and preliminary assessment. For manufacture, procurement, fatigue assessment or safety-critical design, verify the calculation method, dimensions, tolerances, material properties and quality requirements against the applicable current standard and supplier data.
See Also