Page 545 - Shigley's Mechanical Engineering Design

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Mechanical Springs 519

at the inside fiber of the spring. Substitution of τ max = τ, T = FD/2, r = d/2, J =
4
2
πd /32, and A = πd /4 gives
8FD 4F
τ = + (b)
πd 3 πd 2
Now we define the spring index
D
C = (10–1)
d
which is a measure of coil curvature. The preferred value of C ranges from 4 to 12. 1
With this relation, Eq. (b) can be rearranged to give
8FD
τ = K s (10–2)
πd 3
where K s is a shear stress-correction factor and is defined by the equation
2C + 1
K s = (10–3)
2C
The use of square or rectangular wire is not recommended for springs unless
space limitations make it necessary. Springs of special wire shapes are not made in
large quantities, unlike those of round wire; they have not had the benefit of refining
development and hence may not be as strong as springs made from round wire. When
space is severely limited, the use of nested round-wire springs should always be con-
sidered. They may have an economical advantage over the special-section springs, as
well as a strength advantage.

10–2 The Curvature Effect
Equation (10–2) is based on the wire being straight. However, the curvature of the wire
increases the stress on the inside of the spring but decreases it only slightly on the out-
side. This curvature stress is primarily important in fatigue because the loads are lower
and there is no opportunity for localized yielding. For static loading, these stresses can
normally be neglected because of strain-strengthening with the first application of load.
Unfortunately, it is necessary to find the curvature factor in a roundabout way. The
reason for this is that the published equations also include the effect of the direct shear
stress. Suppose K s in Eq. (10–2) is replaced by another K factor, which corrects for
both curvature and direct shear. Then this factor is given by either of the equations
4C − 1 0.615
K W = + (10–4)
4C − 4 C
4C + 2
K B = (10–5)
4C − 3
The first of these is called the Wahl factor, and the second, the Bergsträsser factor. 2
Since the results of these two equations differ by the order of 1 percent, Eq. (10–6)
is preferred. The curvature correction factor can now be obtained by canceling out the

1 Design Handbook: Engineering Guide to Spring Design, Associated Spring-Barnes Group Inc.,
Bristol, CT, 1987.
2 Cyril Samónov, “Some Aspects of Design of Helical Compression Springs,” Int. Symp. Design and
Synthesis, Tokyo, 1984.