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# Basic calculation of Belleville springs

The calculation is carried out in metric or English units. With the ANSI standard set, the calculation is performed in English units.

The calculation is done in XY and XZ planes and their quadratic sum is made.

Calculated Parameters

Unloaded height of truncated cone of free spring

h = H - t [mm, in]

where:

 H unloaded spring height [mm, in] t spring material thickness [mm, in]

Diameter ratio

where:

 D outer spring diameter [mm, in] d inner spring diameter [mm, in]

Calculation factor a

Calculation factorb

Calculation factorg

Limit washer deflection

sm = h [mm, in]

where:

 h unloaded height of truncated cone of free spring [mm, in]

Force at maximum spring deflection (at limit deflection)

where:

 E spring modulus of elasticity [MPa, psi] t spring material thickness [mm, in] sm limit spring deflection [mm, in] m Poisson's ratio [-] a calculation factor [-] D outside spring diameter [mm, in]

Force exerted by the spring at the s deflection

where:

 E spring modulus of elasticity [MPa, psi] t spring material thickness [mm, in] s working deflection of a spring [mm, in] m Poisson's ratio [-] a calculation factor [-] D outside spring diameter [mm, in] h unloaded height of truncated cone of free spring [mm, in]

Maximum pressure stress in the spring at the s deflection

where:

 E spring modulus of elasticity [MPa, psi] t spring material thickness [mm, in] s working deflection of a spring [mm, in] m Poisson's ratio [-] a calculation factor [-] D outside spring diameter [mm, in] h unloaded height of truncated cone of free spring [mm, in] g calculation factor [-]

Total spring number in a set

c = n i [-]

where:

 n spring number in a set with unidirectional mounting [-] i spring number in a set with opposite mounting or number of sets with unidirectional mounting in a set with combined mounting [-]

Stroke (deflection) of a spring set

z = i s [mm, in]

where:

 i spring number in a set with opposite mounting or number of sets with unidirectional mounting in a set with combined mounting [-] s working deflection of a spring [mm, in]

Force exerted by a spring set

F = n F1 [N, lb]

where:

 n spring number in a set with unidirectional mounting [-] F1 force exerted by one washer [N, lb]

Length of spring set in unloaded state

L0 = i (h + n t ) [mm, in]

where:

 h unloaded height of truncated cone of free spring [mm, in] i spring number in a set with opposite mounting or number of sets with unidirectional mounting in a set with combined mounting [-] n spring number in a set with unidirectional mounting [-] t spring material thickness [mm, in]