Torsion spring calculator
Calculate the spring rate, spring moment and bending stress of cylindrical torsion (leg) springs to DIN EN 13906-3. Enter wire diameter, mean coil diameter and number of active coils, specify the angle of rotation or the spring moment – the calculator returns the spring rate as moment per degree, the characteristic line and the bending stress check against an allowable bending stress with a traffic-light rating, live with every input.
Torsion spring calculator (DIN EN 13906-3)
Model: cylindrical torsion (leg) spring made of round wire to DIN EN 13906-3, loaded in bending, static bending stress check with curvature correction q = (w+0.07)/(w-0.75). No fatigue check; the legs (load introduction) are not evaluated and must be verified separately. A sizing tool, not a substitute for a verified spring design.
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Calculating …
Formulas and fundamentals
Unlike compression and extension springs, the wire of a torsion (leg) spring is loaded in bending, not in torsion. The governing modulus is therefore the Young's modulus E, not the shear modulus. The spring rate is usually given as moment per degree: R_M = E·d⁴/(3667·Dm·n) with wire diameter d, mean coil diameter Dm and number of active coils n. The constant 3667 follows from 64·180/pi and links the spring law phi = M·L/(E·I) (with wire length L = pi·Dm·n and second moment of area I = pi·d⁴/64) to the angle in degrees. The spring moment at the working point is M = R_M·phi.
The bending stress in the wire follows from the spring moment and the section modulus of the circular cross section as sigma = 32·M/(pi·d³). Because of the wire curvature the stress on the inner side of the coil is higher than this nominal value; this is captured by the curvature correction factor q, which depends on the spring index w = Dm/d. The calculator uses the documented approximation q = (w+0.07)/(w-0.75). The corrected stress sigma_k = q·sigma governs the strength check.
The static check compares the corrected bending stress sigma_k with an allowable bending stress sigma_allow. This depends on the material and wire diameter and is usually derived from the minimum tensile strength Rm of the spring wire (e.g. EN 10270); for bending it is higher than the allowable shear stress of compression and extension springs. A recommended index range is 4 ≤ w ≤ 20 – outside it the calculator issues a warning. For fluctuating loads a fatigue check is required, and the legs (load introduction) must be verified separately because of additional stresses.
Worked example
A torsion spring made of spring steel (E = 206,000 N/mm²) has a wire diameter d = 3 mm, mean coil diameter Dm = 20 mm and n = 5 active coils. The spring index is w = Dm/d = 6.67, the spring rate R_M = 206,000·3⁴/(3667·20·5) = 45.5 Nmm/degree.
An angle of rotation of phi = 30° produces the spring moment M = R_M·phi = 1365 Nmm. The nominal bending stress is sigma = 32·1365/(pi·3³) = 514.9 N/mm². With the curvature correction q = (6.67+0.07)/(6.67-0.75) = 1.139 the governing stress is sigma_k = 586.4 N/mm².
Against an allowable bending stress of 1000 N/mm² the utilisation is about 59 percent – the static check is satisfied (green). For fluctuating loads a fatigue check would also be required and the legs assessed separately.
Frequently asked questions
How does the torsion spring differ from compression and extension springs?
The torsion (leg) spring is loaded by a moment about its axis, so the wire is stressed in bending instead of torsion. The governing quantity is therefore the Young's modulus E and the bending stress sigma = 32·M/(pi·d³), not the shear modulus and shear stress. The spring rate is given as moment per angle, commonly Nmm per degree.
Why is the spring rate given as moment per degree?
Because the torsion spring travels through an angle, not a length. R_M = E·d⁴/(3667·Dm·n) directly yields the moment per degree of rotation. The constant 3667 (= 64·180/pi) contains the conversion from radians to degrees; working in radians the constant is 64.
What does the curvature correction factor q represent?
The wire is curved into a coil, so the bending stress on the inner side of the coil is higher than the nominal stress of a straight beam. The factor q = (w+0.07)/(w-0.75) with w = Dm/d increases the nominal stress accordingly. The corrected stress sigma_k = q·sigma always governs the strength check.
How large may the allowable bending stress be?
It depends on the material and wire diameter and is derived from the minimum tensile strength Rm of the spring wire (e.g. patented drawn spring steel wire to EN 10270-1). Because the wire is loaded in bending, the allowable static bending stress is higher than the allowable shear stress of compression and extension springs. Governing limit values are taken from the material data sheet or the standard.
Why does the spring index w = Dm/d matter?
A small w means a strongly curved wire with high stress concentration on the inner side and difficult manufacture; a very large w leads to unstable, sensitive springs. The proven range is 4 ≤ w ≤ 20, preferably about 7 to 10. Outside this range the calculator points it out with a note.
Are the legs and fatigue included?
No. The calculator evaluates the static bending stress in the spring body. The legs (load introduction) experience additional bending and notch stresses depending on their shape and support and must be verified separately. For fluctuating loads a fatigue check for body and legs is also required.
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