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]
µ 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]
µ = 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]
=
limit spring deflection [mm, in]
= Poisson's
ratio 
=
calculation coefficient
=
calculation coefficient
= calculation
coefficient 
= calculation
coefficient 
