Spaenaur

Disc Spring Load and Stress Calculations
D
Nomenclature
O.D. = Maximum outside dia. (upper surface)
I.D. = Minimum inside dia. (bottom surface)
h = Conical disc height (cone height)
O.H. = Overall height = Y + h
t = Actual thickness of disc
ß = Cone angle of disc
R = Radius from centreline to load bearing circle
(bottom surface)
M = Ratio factor
µ = Poisson’s ratio (.3 for steel)
E = Young’s modulus (30,000,000 for steel)
f = Deflection of disc
∂ = Ratio of diameters (O.D./I.D.)
P = Load in lbs. at a given deflection
P f = Load in lbs. at flat
X = Sinß • t
Y = Cosß • t
The load-deflection formula was developed by J. Almen
and A. Laszlo, and published in the Transactions of Amer.
Soc. of Mech. Engineers, May 1936, and is rendered as
follows:
LOAD IN LBS. AT A GIVEN DEFLECTION
P = E • f • [(h-f/2)•(h-f) • t+t 3 ]
(1-µ 2 ) • M • R 2
WHERE M = 6 • (∂-1)2
π • L n
∂ ∂ 2
DISC SPRING AT FLAT:
In the flattened condition, the deflection f is equal to
the conical height h and the equation becomes:
E • h • t
P f =
3
(1-µ 2 ) • M • R 2
SIMPLIFIED PROCEDURE
FOR APPROXIMATE LOAD CALCULATIONS
In the flattened condition the load formula is as follows:
P f =
E • h • t 3 (1)
(1-µ 2 ) • M • R 2
By simplification:
K = E (2)
(1-µ 2 ) • M • R 2
Where the K factor is dependent only on the diameters
and the material.
Hence:
P f = K • h • t 3 (3)
For a specific disc spring curvature c = h/t and h = c • t.
The formula becomes by simplification:
P f = K • C • t 4 (4)
By solving this equation for t (thickness), we obtain:
t = 4 P f (5)
K • C
To find the load for any deflection, multiply the load at flat
by a factor I, found in Table 1.
P = P f • I (6)
With the above formulas we have a simple procedure for
determining the load at different deflections or calculating
the thickness for a given load:
1. Find value of constant M in Table 2
2. Solve for constant K
3. Choose C from Table 1
4. If load is given, solve for t (equation 5)
5. If thickness is given, solve for load (equation 4)
6. To find the load for different deflections (equation 6)
Also available
Series AK Disc
Springs for use with
BALL BEARINGS,
page D53.
A well designed disc spring has radii at all corners to
reduce stress concentrations at the edges. A suitable
radius is approx. = t/6. This radius further reduces
dimension R (see Fig. 4).
Usually the overall height of the disc spring is specified
because it is easy to measure and control. The
cone height h, on the other hand, is difficult
to measure (see Fig. 5).
For an approximate calculation, h= (overall height - t)
is acceptable. However, this is not accurate. In fact, h =
(overall height - Y), where Y = Cosß • t. For small
thicknesses (under 2 mm), this is not significant. With
thicker disc springs, this becomes a major factor for
accurate load and stress calculations. This has not been
adequately considered in previous technical literature.
Disc springs 7.49 mm and thicker are made with a bearing
flat at Upper I.D. and Lower O.D. as standard (see Fig. 6).
This bearing flat assures more uniform loading and better
alignment of the disc stack.
The flat is equal approx. to O.D./150. For load calculations,
R must be calculated to the inner edge of the flat.
DISC SPRING STRESS CALCULATIONS
S1 = E • f • [C1 • (h - f/2) + C2 • t]
(1-µ2) • M • R 2
S2 = E • f • [C1 • (h-f/2) - C2 • t]
(1-µ 2 ) • M • R 2
S3 = E • f • [T1 • (h-f/2) + T2 • t]
(1-µ 2 ) • R 2
Where M, C 1
and C 2
are from Table 2, E and µ from Table
3, and
(∂ • L n
∂) - (∂-1) • ∂
T1 =
L n
∂ (∂-1) 2
(.5) • ∂
T2 =
∂-1
∂ = D/d and L n
= natural logarithm. Stress as given is psi
To calculate the load accurately, the following important
factors must be considered:
Disc with theoretical sharp corners. If the disc spring is
made as in Fig. 2, which is unusual, then R = O.D./2. Most
disc springs are made as in Fig. 3.
Therefore, the load bearing radius is not equal to half of
the maximum outside diameter. To calculate R, the angle
B first has to be determined.
.
For evaluation of compressive stress, use formula S1. It
computes the compressive stress at the upper inner
diameter. This compressive stress may be as high as
400,000 psi for certain bolted applications.
For dynamic applications, it is necessary to consider the
tensile stresses at the points marked S2 and S3. The
stresses at these points depend on the ratio of diameters
(∂) and the spring characteristic (C) as well as on the
deflection (f). This stress should not exceed 200,000 psi
at .75h deflection.
SUMMARY
Precise load and stress calculations require the
determination of the disc spring angle ß. Since this is not
easily determined by physical measurement, we have
developed a computer program that calculates the precise
angle and arrives at the exact dimension for conical
height h. This then determines accurate load and stress
calculation. When designing special disc springs and
wishing to evaluate the resultant load and stress with
accuracy, please consult our Engineering Department.
The load and stress formulas are correct only with the
assumption that the spring will be worked within the
elastic limit of the material.
Table 1
To find the load at any intermediate point (between 10%
h and flat), multiply the load at flat by the constant I found
in Table 1 below.
C
Deflection in Percent of h
h/t 10 20 30 40 50 60 70 75 80 90
0.30 0.11 0.21 0.32 0.42 0.52 0.62 0.71 0.76 0.81 0.91
0.40 0.11 0.22 0.33 0.43 0.53 0.63 0.72 0.77 0.82 0.91
0.50 0.12 0.24 0.35 0.45 0.55 0.64 0.73 0.78 0.82 0.91
0.60 0.13 0.25 0.36 0.47 0.57 0.66 0.75 0.79 0.84 0.92
0.70 0.14 0.27 0.39 0.49 0.59 0.68 0.77 0.81 0.85 0.92
0.80 0.16 0.29 0.41 0.52 0.62 0.71 0.79 0.83 0.86 0.93
0.90 0.17 0.32 0.45 0.56 0.65 0.74 0.81 0.85 0.88 0.94
1.00 0.19 0.34 0.48 0.59 0.69 0.77 0.84 0.87 0.90 0.95
1.05 0.19 0.36 0.50 0.61 0.71 0.79 0.85 0.88 0.91 0.96
1.10 0.20 0.37 0.52 0.63 0.73 0.80 0.87 0.89 0.92 0.96
1.15 0.21 0.39 0.54 0.65 0.75 0.82 0.88 0.91 0.93 0.97
1.20 0.22 0.41 0.56 0.68 0.77 0.84 0.90 0.92 0.94 0.97
1.25 0.23 0.43 0.58 0.70 0.79 0.86 0.91 0.93 0.95 0.98
1.30 0.25 0.44 0.60 0.73 0.82 0.88 0.93 0.95 0.96 0.98
1.35 0.26 0.46 0.63 0.75 0.84 0.91 0.95 0.96 0.98 0.99
1.40 0.27 0.48 0.65 0.78 0.87 0.93 0.97 0.98 0.99 1.00
1.50 0.29 0.52 0.70 0.83 0.92 0.98 1.01 1.01 1.02 1.01
1.60 0.32 0.57 0.76 0.89 0.98 1.03 1.05 1.05 1.05 1.03
1.80 0.38 0.67 0.88 1.02 1.12 1.14 1.14 1.13 1.11 1.06
2.00 0.44 0.78 1.01 1.17 1.25 1.27 1.25 1.22 1.18 1.10
2.50 0.63 1.10 1.40 1.60 1.67 1.65 1.55 1.48 1.40 1.21
3.00 0.87 1.48 1.91 2.13 2.19 2.11 1.93 1.81 1.66 1.35
3.50 1.15 1.96 2.49 2.75 2.80 2.66 2.37 2.19 1.98 1.51
4.00 1.47 2.50 3.16 3.50 3.50 3.29 2.88 2.63 2.34 1.69
Table 2
Constant M, C 1
and C 2
∂
∂
OD/ID M C 1
C 2
OD/ID M C 1
C 2
1.10 .166 .986 1.002 2.10 .706 1.242 1.416
1.15 .232 1.001 1.025 2.20 .721 1.264 1.453
1.20 .291 1.016 1.048 2.30 .733 1.286 1.490
1.25 .342 1.030 1.070 2.40 .742 1.307 1.527
1.30 .388 1.044 1.092 2.50 .750 1.328 1.563
1.35 .428 1.058 1.114 2.60 .757 1.348 1.599
1.40 .463 1.072 1.135 2.80 .767 1.388 1.669
1.45 .495 1.085 1.157 3.00 .773 1.426 1.738
1.50 .523 1.098 1.178 3.20 .776 1.464 1.806
1.60 .571 1.124 1.219 3.40 .778 1.500 1.873
1.70 .610 1.149 1.260 3.60 .778 1.535 1.938
1.80 .642 1.173 1.300 3.80 .777 1.570 2.003
1.90 .668 1.197 1.339 4.00 .775 1.604 2.067
2.00 .689 1.220 1.378
Table 3
Modulus of elasticity and Poisson’s ratio for
different materials
E Modulus Vs. Temperature in F° Poisson’s
Material 68F° 250F° 400F° 600F° Ratio µ
Steel - 1075 30 x 10 6 29.5 x 10 6 — — 0.30
Steel - 6150 30 x 10 6 29.8 x 10 6 28.5 x 10 6 — 0.30
Stainless 17/7 PH 29 x 10 6 N/A N/A 26.5 x 10 6 0.34
Stainless 302 28 x 10 6 N/A 26.5 x 10 6 — 0.30
Inconel x -750 31 x 10 6 30.8 x 10 6 29.5 x 10 6 28.3 x 10 6 0.29
CATALOG 13
D44