Stress in Helical Springs


Details

Fig1 shows a round-wire helical compression spring loaded by the axial force F. D is designated as the mean coil diameter and d as the wire diameter. Now imagine that the spring is cut at some point, a portion of it removed, and the effect of the removed portion replaced by the net internal reactions as shown in Fig2. Then from equilibrium the cut portion would contain a direct shear force F and a torsion T = F D/2.

To visualize the torsion, picture a coiled garden hose. Now pull one end of the hose in a straight line perpendicular to the plane of the coil. As each turn of hose is pulled off the coil, the hose twists or turns about its own axis. The flexing of a helical spring creates a torsion in the wire in a similar manner.


Fig 1.

Fig. 2

The maximum stress in the wire may be computed by superposition of the direct shear stress given by Eq2 of the lesson Uniformly Distributed Stress, and the torsional shear stress by Eq3 of the lesson Torsion. Substituting F for P and r for rmax, respectively, the equation is then:
(Eq1)    
τmax =
Tr
J
 + 
F
A

at the inside fiber of the spring. Substitution of τmax = τ, T = FD/2, r = d/2, J = πd4/32, and A = πd2/4 gives:

(Eq2)    
τ =
8FD
πd3
 + 
4F
πd2

Now we define the spring index:

(Eq3)    
C =
D
d

which is a measure of coil curvature. With this relation, Eq2 can be rearranged to give:

(Eq4)    
τ = Ks
8FD
πd2

where Ks is a shear-stress correction factor and is defined by the equation:

(Eq5)    
Ks =
2C + 1
2C

For most springs, C ranges from about 6 to 12. Eq4 is quite general and applies for both static and dynamic loads.

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 considered. They may have an economical advantage over the special-section springs, as well as a strength advantage.