### 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:**

Preview: **Force at Maximum Spring Deflection and Limit Deflection Calculator**

Equations used in above calculator

Where:

E = Spring modulus of elasticity [Pa, psi]

t = Spring material thickness [mm, in]

s_{m} = 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

F_{1} = 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**

L_{0} = Height of spring set in unloaded state [mm, in]

z = stroke (deflection) of spring set [mm, in]

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