Page 546 - Shigley's Mechanical Engineering Design

bud29281_ch10_517-568.qxd 12/16/2009 7:14 pm Page 520 pinnacle 203:MHDQ196:bud29281:0073529281:bud29281_pagefiles:

520 Mechanical Engineering Design
effect of the direct shear. Thus, using Eq. (10–5) with Eq. (10–3), the curvature cor-
rection factor is found to be
K B 2C(4C + 2)
K c = = (10–6)
K s (4C − 3)(2C + 1)
Now, K s , K B or K W , and K c are simply stress-correction factors applied multiplica-
tively to Tr/J at the critical location to estimate a particular stress. There is no stress-
concentration factor. In this book we will use
8FD
τ = K B (10–7)
πd 3
to predict the largest shear stress.
10–3 Deflection of Helical Springs

The deflection-force relations are quite easily obtained by using Castigliano’s theorem.
The total strain energy for a helical spring is composed of a torsional component and
a shear component. From Eqs. (4–18) and (4–20), p. 162, the strain energy is
2
2
T l F l
U = + (a)
2GJ 2AG
2
4
Substituting T = FD/2, l = π DN, J = πd /32, and A = πd /4 results in
2
2
3
4F D N 2F DN
U = + (b)
2
4
d G d G
where N = N a = number of active coils. Then using Castigliano’s theorem, Eq. (4–26),
p. 165, to find total deflection y gives
3
∂U 8FD N 4FDN
y = = + (c)
2
4
∂F d G d G
Since C = D/d, Eq. (c) can be rearranged to yield
3
3
8FD N 1 . 8FD N
y = 1 + = (10–8)
4
4
d G 2C 2 d G
The spring rate, also called the scale of the spring, is k = F/y, and so
4
. d G
k = (10–9)
3
8D N
10–4 Compression Springs
The four types of ends generally used for compression springs are illustrated in Fig. 10–2.
A spring with plain ends has a noninterrupted helicoid; the ends are the same as if a
long spring had been cut into sections. A spring with plain ends that are squared or
closed is obtained by deforming the ends to a zero-degree helix angle. Springs should
always be both squared and ground for important applications, because a better transfer
of the load is obtained.
Table 10–1 shows how the type of end used affects the number of coils and the
3
spring length. Note that the digits 0, 1, 2, and 3 appearing in Table 10–1 are often
3 For a thorough discussion and development of these relations, see Cyril Samónov, “Computer-Aided
Design of Helical Compression Springs,” ASME paper No. 80-DET-69, 1980.