Springs Washer Belleville Equation

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Springs Washer Belleville Equation

A Belleville spring or washer is formed into a conical or cone shaped geometry. The slight or aggressive conical shape gives the washer a spring characteristic and action. Belleville washers are typically used as springs where the spring action is used to apply a pre-load or flexible quality to a bolted joint. The equations below will determine the various characteristics including the applied load of a Belleville spring or washer.

Springs Washer Belleville Equation / Formula

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]
d = inner spring diameter [mm]


Calculation coefficient

Calculation coefficient

Calculation coefficient

Limit Washer Deflection

Where:

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


Force at Maximum Spring Deflection and Limit Deflection:

Where:

E = Spring modulus of elasticity [MPa, psi]
t = Spring material thickness [mm, in]
= limit spring deflection [mm, in]
= Poisson's ratio
= calculation coefficient
D = outside spring diameter [mm, in]


Force Exerted by the Spring at s Deflection:

Where:

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


Maximum Pressure Stress in Spring at s Deflection:

Where:

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


Total Springs in a Set or Stack up

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 of Deflection of a Spring Set  (Stack up)

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 (Stack up)

Where:

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


Height of Spring Stack up Unloaded

Where:

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


Height of Loaded Spring Stack up

L0 = Height of spring set in unloaded state [mm, in]
z = stroke (deflection) of spring set [mm, in]

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