Helical Springs

Shear Stress in Springs

For light springs (small $d/4R$ ratio), the shear stress is approximated as:

$$\tau = \frac{16PR}{\pi d^3} \left( 1 + \frac{d}{4R} \right)$$

For heavy springs, where curvature effects are more significant, Wahl’s formula provides a more accurate expression:

$$\tau = \frac{16PR}{\pi d^3} \left( \frac{4m - 1}{4m - 4} + \frac{0.615}{m} \right)$$

Here, $m$ is the spring index, and the factor $\tfrac{4m - 1}{4m - 4}$ is known as the Wahl factor. $m=2R/d\ or\ m=D/d$

Deflection of the Spring

The elongation (deflection) of the spring under load $P$ is:

$$\delta = \frac{64PR^3n}{G d^4}$$

Spring Constant

The ratio of load to deflection defines the spring constant $k$:

$$k = \frac{P}{\delta} = \frac{G d^4}{64 R^3 n} $$