Belleville Spring Washer Equation and Calculator

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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

Springs Washer Belleville Equation / Formula


Force at Maximum Spring Deflection and Limit Deflection:

Preview: Force at Maximum Spring Deflection and Limit Deflection Calculator

Equations used in above calculator
Force at Maximum Spring Deflection and Limit Deflection

Where:

E = Spring modulus of elasticity [Pa, psi]
t = Spring material thickness [mm, in]
sm = Total or limit of spring deflection [mm, in]
µ = Poisson's ratio
α = calculation coefficient (see equation below)
D = outside spring diameter [mm, in]


Force Exerted by the Spring at s Deflection:

Force Exerted by the Spring at s Deflection

Where:

E = spring modulus of elasticity [Pa, psi]
t = spring material thickness [mm, in]
s = Design or working deflection of a spring [mm, in]
µ = Poisson's ratio
α = 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:

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 
α = calculation coefficient 
D = outside spring diameter [mm, in]
β = calculation coefficient 
h = unloaded height of truncated cone of free spring [mm, in]
γ = calculation coefficient 


Diameter Ratio

Diameter Ratio

Where:

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


Calculation coefficient α

Calculation coefficient

Calculation coefficient β

Calculation coefficient β

Calculation coefficient γ

Calculation coefficient γ

Limit Washer Deflection

Limit Washer Deflection

Where:

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

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]


Total Springs in a Set or Stack up

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)

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)

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

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

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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