Belleville Washer Design Formulae and Calculator

Related Resources: calculators

Belleville Washer Design Formulae and Calculator

Spring Design and Engineering Equations and Calculators

Belleville Washer Design Formula and Calculator

When a load is applied to a Belleville washer it tends to flatten causing radial and circumferential strains. This elastic deformation creates the spring action. A typical Belleville washer is shown in Figure 1.

Figure 1 - Typical Belleville Washer

Preview: Belleville Design Calculator

Belleville washers are capable of providing very high loads at small deflections. Stress is not distributed uniformly in Belleville washers. The highest stress occurs at the top inner edge and can be estimated with the following equation:

Eq. 1
S = ( E_M · f · R ) / ( 1 - µ² ) ( t / a² )

Eq. 1a
a = OD / 2

Eq. 1b
R = OD / ID

Where:

S = Bending stress, lbs/in², (N.mm²)
E_M = Modulus of Elasticity, lbs/in², (N.mm²)
µ = Poisson's ratio
f = washer deflection, in (mm)
t = washer thickness, in (mm)
a = radius, in (mm)
OD = outside diameter, in (mm)
ID = Inside diiameter, in (mm)
R = Dimension factor

The failure rate of a curved washer is determined using the following equation:

Eq. 2

λ_SP = λ_SP,B · ( S / T_s )³ · C_cs · C_R · C_M

A generalized equation that adjusts the base failure rate of a curved washer considering anticipated operating conditions can be established:

Eq. 3
λ_SP = λ_SP,B · C_E · C_t · C_D · C_f · C_Y · C_S · C_CS · C_R · C_M

where:

λ_SP = Failure rate of torsion spring, failures/million hours
λ_SP,B = Base failure rate for torsion spring, 14.3 failures/million hours

C_E = Multiplying factor which considers the effect of the material elasticity modulus on the base failure rate
Eq. 4
C_E = ( E_M / 28.5 x 10⁶ )³

C_t = Multiplying factor which considers the effect of the material thickness on the base failure rate
Eq. 5
C_t = ( t / 0.025 )³

C_D = Multiplying factor which considers the effect of washer diameter on the base failure rate
Eq. 6
C_D = ( 1.20 / OD )⁶

C_Y = Multiplying factor which considers the effect of material tensile strength on the base failure rate
Eq. 7
C_Y = ( 190 x 10³ / T_s )³
T_s = Tensile Strength, lbs/in², (N/mm²)

C_f = Multiplying factor which considers the effect of washer deflection on the base failure rate
Eq. 8
C_f = ( f / 0.055 )³

C_NW = Multiplying Factor which considers the number of waves on the base failure rate
Eq. 9
C_NW = ( 5 / NW )²

C_CS = Multiplying factor which considers the effect of a corrosive environment on the base failure rate
Eq. 10a
C_CS = 0.100 If C_R ≤ cycles/min
Eq. 10b
C_CS = C_R / 300 ) For 30 cycles/min < CR ≤ 300 cycles min,
Eq. 10c
C_CS = ( C_R / 300 )³ ForC_R > 300 cycles/min,

C_R = Multiplying factor which considers the effect of a corrosive environment on the base failure rate
Eq. 11
C_R = 1.0 unless or greater than 1.0 with user's experience with the spring and the operating environment.

C_M = Multiplying factor which considers the effect of the manufacturing process on the base failure rate
Eq. 12
C_M = 1.0 a higher value for the multiplying factor is used based on previous experience with the manufacturer.

C_S = Multiplying factor for compressive stress
Eq. 12
C_S = ( 6 / ( π · ln R )³ ) · ( R - 1 / ( ln R) - 1 ) · ( (R - 1 ) / 2 ) · ( ( R - 1 )² / R² )

Reference:

Handbook of of Reliability Predictions Procedure for Mechanical Equipment
Logistics Technology Support
CARDEROCKDIV, NSWC-11
2011