This cheat sheet covers the main formulas and design ideas used for helical compression springs in engineering. Students need it because spring problems combine geometry, material properties, force, deflection, and stress in one system. A clear reference helps organize the steps for sizing a spring and checking whether it can safely carry a load.
Key Facts
- The mean coil diameter is D = OD - d = ID + d, where OD is outside diameter, ID is inside diameter, and d is wire diameter.
- The spring index is C = D / d, and common practical designs often use C between 4 and 12.
- The spring rate for a close-coiled helical compression spring is k = Gd^4 / (8D^3Na), where G is shear modulus and Na is the number of active coils.
- Spring deflection under an axial load is x = F / k, where F is force and k is spring rate.
- Basic torsional shear stress in the wire is tau = 8FD / (pi d^3).
- Corrected maximum shear stress is tau_max = Kw(8FD / (pi d^3)), where Kw accounts for curvature and direct shear effects.
- The Wahl correction factor is Kw = (4C - 1) / (4C - 4) + 0.615 / C.
- For springs in series, 1 / k_total = 1 / k1 + 1 / k2, and for springs in parallel, k_total = k1 + k2.
Vocabulary
- Helical compression spring
- A coil spring designed to shorten under an axial compressive load and return toward its original length when unloaded.
- Spring rate
- The stiffness of a spring, calculated as k = F / x, where F is force and x is deflection.
- Mean coil diameter
- The average diameter of the coil, measured from the centerline of the wire on one side to the centerline on the opposite side.
- Active coils
- The coils that deform significantly when the spring is loaded and therefore contribute to spring deflection.
- Spring index
- The ratio C = D / d, comparing mean coil diameter to wire diameter and indicating how tightly the spring is coiled.
- Wahl factor
- A stress correction factor that adjusts the basic shear stress for wire curvature and direct shear in a helical spring.
Common Mistakes to Avoid
- Using outside diameter instead of mean coil diameter is wrong because the main spring formulas require D measured at the wire centerline.
- Counting all coils as active coils is wrong because end coils may be inactive depending on the end style and do not fully contribute to deflection.
- Forgetting the fourth power on wire diameter in k = Gd^4 / (8D^3Na) is wrong because small changes in wire diameter strongly affect stiffness.
- Ignoring the Wahl factor is wrong because the uncorrected shear stress can underestimate the maximum stress in the spring wire.
- Mixing units such as millimeters, meters, newtons, and pounds in one calculation is wrong because formulas only give valid results when units are consistent.
Practice Questions
- 1 A compression spring has G = 79,000 N/mm^2, d = 4 mm, D = 32 mm, and Na = 8. Calculate the spring rate k.
- 2 A spring with rate k = 12 N/mm is compressed by 25 mm. What force does it exert?
- 3 A spring has d = 5 mm, D = 40 mm, and F = 150 N. Calculate the basic torsional shear stress tau = 8FD / (pi d^3).
- 4 If two springs have the same material, wire diameter, and active coils, but one has a larger mean coil diameter, which spring has the lower spring rate and why?