BELLEVILLE SPRING

A Belleville spring consists of coned disk, as shown in Fig. 01.It resembles to a dinner plate without

bottom. This type of spring is also called coned disk spring. It is also called Belleville spring because it was

invented by Julia Belleville, who patented its design in France in 1867. Belleville spring has typical load

deflection characteristic, as shown in Fig. 02. The variation ofht

ratio produces a wide variety of load

deflection curves. For example, when ht

ratio is 3.5, S curve is obtained which is useful in applications

involving snap acting mechanism. When ht

is reduced to 2.1, the central portion of curve becomes

horizontal, which means that load is constant for this range of deflection. This portion of the curve is useful

for engaging or disengaging the clutch, when Belleville spring is used as a clutch spring. Belleville spring is

offers following advantages:

Fig. 01 Belleville Spring

Fig. 02 Load Deflection Curves for Belleville Springs

It is simple construction and easy to manufacture.

Belleville spring is a compact spring unit.

It is especially useful where very large force is desired for small deflection of spring.

It provides a wide range of spring constants making it versatile.

It can provide any linear or non-linear load deflection characteristic.

The individual coned disks of a particular size and thickness can be stacked in series, parallel or

series parallel combinations, as shown in Fig. 03. These combinations provide a variety of spring

constants without changing the design. When two Belleville springs are arranged in series, double

deflection is obtained for the same force. On the other hand, when two Belleville springs are in

parallel, almost double fore is obtained for a given deflection.

Nesting of Belleville Springs

Series Combination

Parallel Combination

Parallel Series Combination

Belleville springs are used in plate clutches and brakes, gun recoil mechanisms, relief valves

and a wide variety of bolted connections .

The analysis of Belleville spring is exceedingly complex and mathematical treatment is beyond

the scope of the topic. The load-deflection and load-stress equations of Belleville spring are as

follows:

P=Eδ

(1−μ2)M ¿¿

σ=Eδ

(1−μ2 )M ¿¿

Where:

P = Axial force (N)

δ= Deflection spring (m)

t= Thickness of washer (m)

h= Free height minus thickness (m)

E= Modulus of elasticity ( N/m2)

σ= Stress at the inside circumference ( pa)

do= Outer diameter of washer (m)

d i= Inner diameter of washer (m)

μ= Poisson’s ratio (0.3) for steel

M=6

π loge (do /di )¿¿

C1=¿ 6

π log e(dodi )[ (do /d i)−1loge (d o /di )

−1]¿

C2=¿ 6

πlog e(do /di) [ (do /d i)−12 ]¿

PROBLEM:

A Belleville spring is made of silicon steel. The spring is compressed completely flat when

it is subjected to axial force of 4,200N. The corresponding maximum stress is (1375MPa). Assume,

dod i

= 1.75 andht

= 1.5

Calculate:

thickness of washer ;

free height of washer minus thickness (h) ;

outer diameter of washer ; and

inner diameter of washer.

SOLUTION:

When spring is compressed completely flat,δ = h

We used our formula:

M=6

π loge (do /di )¿¿

M= 6π loge (1.75 ) [ (1.75)−1(1.75) ]

2

= 0.6268

C1=¿ 6

π log e(dodi )[ (do /d i)−1loge (d o /di )

−1]¿

C1=¿

6π loge(1.75) [ (1.75 )−1

loge (1.75)−1]¿= 1.161

C2=¿ 6

πlog e(do /di) [ (do /d i)−12 ]¿

C2=¿6

πlog e(1.75) [ (1.75 )−12 ]¿ = 1.28

Dividing pσ

pσ

= Eδ

(1−μ2)M ¿¿¿

pσ

=[ (h−δ /2 ) (h−δ ) t+t 3 ][C1 (h−δ /2¿+C2t ) ]

Since: h =δ

pσ

= t 3

C1 (h−δ /2¿+C2t )

Substituting the value;

4200N1375MPa

= t3

1.161 (1.5 t−1.5 t /2¿+1.28 t )

4200N1375MPa

= t 2

2.15

t =2.5630219 x 10−3 m or 2.563 mmh= 1.5(2.5630219 x 10−3m)

h =3.844532921x 10−3m or 3.8445 mm or 3.9 mmSolve for inner and outer diameter:

P=Eδ

(1−μ2)M ¿¿

Since h=δ

P=Eδ

(1−μ2)M ¿¿(t 3¿

E= 207,000 MPa

μ=0.3

t= 2.5630219 x 10−3m

h= 3.844532921x 10−3m

Substituting the value,

4200N=(207,000MPa∗0.0039)

(1−.32 ) .6268¿¿(.0025633 ¿

do=0. 1495688596m or 149.5688596mmd i=

do1.75

d i=149.5688596mm

1.75

d i=85.46791977mm