Helical compression springs
How spring rate depends on wire diameter, coil diameter, active turns and material shear modulus.
ENGINEERING PRINCIPLE
A close-coiled round-wire spring stores energy mainly through torsion of its wire; wire diameter has a strong fourth-power effect on rate.
k = Gd⁴/(8D³N)Variables and units
- k = spring rate (N/mm)
- G = shear modulus (N/mm²)
- d = wire diameter (mm)
- D = mean coil diameter (mm)
- N = active coils
Worked example
For G = 79 GPa, d = 5 mm, D = 35 mm and N = 8, k is approximately 17.99 N/mm.
Calculation method
- Convert G from GPa to N/mm².
- Use the mean coil diameter and active, not total, coils.
- Calculate rate, then evaluate stress and available deflection separately.
Practical applications
- Return springs
- Valve and actuator springs
- Isolation and preload mechanisms
Review boundary and reference
- Solid height, buckling, end condition, fatigue and surge require additional checks.
- A selected wire material needs condition-specific properties.
Formula and displayed units independently checked for the stated conditions. This is not a code-compliance design method.
Source: J. E. Shigley et al., Mechanical Engineering Design, close-coiled helical compression springs.