Spring rate, deflection under load and Wahl-corrected torsional shear stress with safety factor — plus free length, solid height, clash clearance, spring index and a buckling slenderness check.
A helical spring works by twisting its wire. Rate depends on wire diameter to the fourth power and coil diameter to the third, which is why small dimensional changes shift stiffness so sharply. Stress is torsional, corrected by the Wahl factor for wire curvature and direct shear.
dw wire diameter (mm) ·
D mean coil diameter (mm) ·
Na active coils ·
Nt total coils ·
F load (N) ·
G shear modulus (MPa) ·
Ssy allowable shear stress (MPa)
A hard-drawn steel spring, 4 mm wire on a 25 mm mean coil diameter, 10 active coils, closed and ground ends, carrying 80 N.
The spring is lightly stressed at SF 7.09 and has 5 mm of travel left before the coils touch. Doubling the load to 160 N would halve the safety factor to 3.54 and consume 9.89 mm of the 10 mm available travel — leaving only 0.11 mm reserve, which is too tight for production.
| Quantity | Symbol | Unit | Accepted range |
|---|---|---|---|
| Wire diameter | dw | mm | > 0, less than D |
| Mean coil diameter | D | mm | > dw |
| Active coils | Na | — | ≥ 1 |
| Applied load | F | N | > 0 |
| Shear modulus | G | MPa | > 0 |
| Allowable shear | Ssy | MPa | > 0 |
| Spring rate | k | N/mm | output |
| Shear stress | τ | MPa | output |
| Spring index | C | — | output, target 4–12 |
| Material | G (MPa) | Ssy (MPa) | Typical use |
|---|---|---|---|
| Hard-drawn steel | 79 000 | 700 | General purpose, lowest cost |
| Heat-treated steel | 79 000 | 550 | Formed then heat treated |
| Stainless 302 | 69 000 | 480 | Corrosion and food service |
| Chrome silicon | 77 200 | 750 | High stress, shock and fatigue duty |
| Ends | Total coils Nt | Free length Lf | Notes |
|---|---|---|---|
| Closed and ground | Na + 2 | 1.25·Na·dw + 2·dw | Squarest seating, most common |
| Closed, not ground | Na + 2 | 1.25·Na·dw + 3·dw | Cheaper, less square |
| Open (plain) | Na | 1.25·Na·dw + dw | Poor seating, light duty only |
Spring rate equals the shear modulus × wire diameter⁴, divided by 8 × mean coil diameter³ × active coils. Because wire diameter enters to the fourth power and coil diameter to the third, small changes in either have a large effect. Increasing wire diameter by 10 % stiffens the spring by about 46 %, while increasing coil diameter by 10 % softens it by about 25 %.
Spring index is the mean coil diameter divided by the wire diameter. It is the single best indicator of whether a spring can be made economically. Below about 4 the wire is bent so tightly that coiling becomes difficult and residual stresses are high. Above about 12 the spring becomes floppy and hard to handle, and tolerances suffer. Most production springs sit between 5 and 9.
The simple torsion formula underestimates stress in a helical spring because the wire is curved, which concentrates stress on the inside of the coil, and because direct shear adds to the torsional shear. The Wahl factor corrects for both. It grows as spring index falls, so tightly wound springs with a low index carry a significantly higher real stress than the uncorrected calculation suggests.
Solid height is the length of the spring when every coil touches, equal to total coils × wire diameter. Clash allowance is the gap remaining between the working length and solid height. If a spring reaches solid under load the rate becomes effectively infinite and the load spikes, which can damage the spring or the surrounding parts. A common target is to keep at least 10–15 % of the working deflection in reserve.
Buckling risk is governed by slenderness, the free length divided by the mean coil diameter. Above about 4 a spring compressed between flat parallel ends becomes prone to sideways buckling, much like a slender column. The remedy is a guide rod through the centre, a bore around the outside, or a redesign with a larger coil diameter and fewer coils.