Spring rate
-Deflection at M
-Bending stress at M
-Coil OD when loaded
-All results
| Mean coil diameter D | - |
| Spring index C = D / d | - |
| Stress factor Ki | - |
| Active coils Na | - |
| Young's modulus E | - |
| Wire tensile strength (estimate) | - |
| Allowable static stress | - |
| Maximum static torque | - |
| Bending stress / allowable | - |
Method and formulas
Standard helical torsion spring theory for round wire, so every figure can be checked by hand.
| Quantity | Formula |
|---|---|
| Mean diameter and index | D = OD − d, C = D / d |
| Active coils | Na = Nb + (L1 + L2) / (3πD) |
| Rate per turn | k = E d4 / (10.8 D Na); per degree = k / 360 |
| Deflection | θ = 360 M / k degrees |
| Stress factor | Ki = (4C2 − C − 1) / (4C (C − 1)) |
| Bending stress | σ = Ki · 32 M / (π d3) |
| Wire tensile strength | Sut = A / dm (d in mm), A and m by material |
| Allowable static stress | 78 % of Sut (61 % for stainless) |
| Coil diameter when wound | D' = Nb D / (Nb + θ/360), OD' = D' + d |
Material constants (E; A and m) are commonly published values: music wire E 203 GPa, A 2211 MPa mmm, m 0.145; hard drawn 196 GPa, 1783, 0.190; oil tempered 196 GPa, 1855, 0.187; stainless 193 GPa, 1867, 0.146. They are estimates for wire in the middle of its size range. The factor 10.8 includes allowance for friction on the arbor.
Worked example: music wire, 2.0 mm wire, 16 mm OD, 6 body coils, two 25 mm arms, 500 N-mm torque
D = 16 − 2 = 14 mm, C = 7.0. Na = 6 + 50 / (3π × 14) = 6.379. k = 203 000 × 24 / (10.8 × 14 × 6.379) = 3368 N-mm per turn (9.35 N-mm per degree). θ = 53.5 degrees. Ki = 1.1190, so σ = 1.1190 × 32 × 500 / (π × 23) = 712 MPa. Sut = 2211 / 20.145 = 2000 MPa, allowable 78 % = 1560 MPa, so the spring is at 46 % of the allowable. Maximum static torque 1095 N-mm. Coil diameter when wound D' = 13.66 mm.
Questions
How do you calculate torsion spring rate?
The rate per turn is k = E d^4 / (10.8 D Na) in torque per turn, where E is the Young's modulus of the wire, d the wire diameter, D the mean coil diameter and Na the active coils. The factor 10.8 is the usual practical value; pure beam theory gives 10.186 and the difference allows for friction between the coils and the arbor. Divide by 360 to get the rate per degree.
What are the active coils of a torsion spring?
Na = Nb + (L1 + L2) / (3 pi D), where Nb is the number of body coils and L1, L2 are the lengths of the two straight arms. The arms bend as cantilevers and add flexibility equal to a fraction of a turn.
What stress is used for a torsion spring?
The wire is loaded in bending, not torsion: sigma = Ki x 32 M / (pi d^3), with the inner-fibre stress factor Ki = (4C^2 - C - 1) / (4C (C - 1)) and spring index C = D / d. The result is compared with a fraction of the wire tensile strength; about 78 percent of the tensile strength is a common static guide for stress-relieved carbon wire and about 61 percent for stainless.
Does a torsion spring get smaller when it is loaded?
Yes. When wound in the direction that tightens the coils the body diameter shrinks to D' = Nb D / (Nb + deflection in turns). Leave clearance between the coil and the arbor or the spring will bind and the rate will rise.
Which way should a torsion spring be loaded?
Load it in the direction that winds the coil tighter. The residual stresses from coiling then oppose the working stress. Loading it to unwind the coil raises the stress and shortens life.
What spring index is a good choice?
An index C between about 4 and 12 is practical. Below 4 the wire is hard to coil and the stress factor is high; above 12 the spring is floppy and hard to hold to tolerance.
What does this calculator not cover?
It covers the coil body and the arm allowance: rate, deflection, bending stress, coil diameter and a static torque limit. It does not check arm bending stresses, fatigue life, relaxation, temperature or the effect of the arbor. Use the spring maker's data for cycled or safety-critical springs. Verify critical designs.
Engineering aid for guidance. It checks the coil body and arm allowance only and does not cover arm stresses, fatigue, relaxation, temperature or arbor effects. Verify critical designs.