### Belleville Spring Washer Equation and Calculator

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  Force at Maximum Spring Deflection and Limit Deflection:

Equations used in above calculator 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: 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: 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 Where:

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

Calculation coefficient α Calculation coefficient β Calculation coefficient γ 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 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]