Calculate moment rate, moments and bending stresses of a cylindrical helical torsion spring made of round wire to EN 13906-3.
Set the inputs and run the calculation.
Determine moment rate, torque and bending stress of a helical torsion spring from wire diameter, coil diameter, coil count and two rotation angles.
Wire diameter d, mean coil diameter D, number of active coils n, and two rotation angles α₁ and α₂ relative to the unloaded position are required; plus whether the load opens or closes the coils.
A torsion spring's highest bending stress is on the inside of the wire. Leg angle and support define the real moment because friction or contact can change the ideal free characteristic.
The moment rate follows from RM = E*d^4/(3667*D*n); the moment at each angle is M(alpha) = RM*alpha. The nominal bending stress sigma = 32*M/(pi*d^3) is corrected for opening loads with the curvature factor q = (w+0.07)/(w-0.75), where w = D/d is the spring index.
d = 2 mm, D = 16 mm (w = 8), n = 6, material EN 10270-1 (E = 206,000 MPa), alpha_1 = 20 deg, alpha_2 = 90 deg: RM = 206,000*2^4/(3667*16*6) = 9.363 N*mm/deg. M(alpha_2) = 9.363*90 = 842.7 N*mm. sigma(alpha_2) = 32*842.7/(pi*2^3) = 1072.9 MPa, corrected with q = (8+0.07)/(8-0.75) = 1.113 to sigma_k(alpha_2) = 1194.2 MPa.
Sources and limits: DIN EN 13906-3:2014-06, moment-rate, moment and bending-stress equations with curvature correction (notation follows the standard).
Determine moment rate, torque and bending stress of a helical torsion spring from wire diameter, coil diameter, coil count and two rotation angles.
A helical torsion spring is rotated about its coiling axis through its legs and stores energy as bending of the coiled wire -- not torsion, despite the name. The highest stress occurs at the inside fibre of the wire, where bending stress and a curvature effect superimpose; that effect is larger for opening loads (coils widen) than for closing loads. This calculator represents only the spring body: wire diameter d, mean coil diameter D, coil count n and the two rotation angles alpha_1/alpha_2 determine moment rate, moment and bending stress -- leg bending, supports and friction are separate checks.
RM = E*d^4 / (3667*D*n)
M(alpha) = RM*alphasigma = 32*M / (pi*d^3)q = (w + 0.07) / (w - 0.75), w = D/dsigma_k = q*sigma (opening load only)| Symbol / input | Meaning |
|---|---|
| RM | Spring moment rate in N*mm per degree of rotation. |
| E | Young's modulus of the spring wire, e.g. 206,000 MPa for EN 10270-1/-2, 180,000 MPa for EN 10270-3. |
| d | Diameter of the round spring wire in mm. |
| D | Mean coil diameter (wire centreline to centreline) in mm. |
| n | Number of active spring coils. |
| alpha | Rotation angle relative to the unloaded position, in degrees. |
| M(alpha) | Spring moment at the given rotation angle in N*mm. |
| sigma | Nominal bending stress without curvature correction, in MPa. |
| w | Spring index w = D/d, valid in this calculator for 4 <= w <= 20. |
| q | Bending-stress curvature factor per EN 13906-3, applied for an opening load. |
Wire diameter d, mean coil diameter D, number of active coils n, and two rotation angles α₁ and α₂ relative to the unloaded position are required; plus whether the load opens or closes the coils. A torsion spring's highest bending stress is on the inside of the wire. Leg angle and support define the real moment because friction or contact can change the ideal free characteristic.
Enter wire diameter d, mean coil diameter D, number of active coils n and the two rotation angles alpha_1 and alpha_2. State whether the load opens or closes the coils -- this decides whether the curvature factor is applied.
d = 2 mm, D = 16 mm (w = 8), n = 6, material EN 10270-1 (E = 206,000 MPa), alpha_1 = 20 deg, alpha_2 = 90 deg: RM = 206,000*2^4/(3667*16*6) = 9.363 N*mm/deg. M(alpha_2) = 9.363*90 = 842.7 N*mm. sigma(alpha_2) = 32*842.7/(pi*2^3) = 1072.9 MPa, corrected with q = (8+0.07)/(8-0.75) = 1.113 to sigma_k(alpha_2) = 1194.2 MPa.
The moment rate RM is constant over the linear range -- moment M(alpha) grows proportionally with rotation angle, much as a compression spring's force grows proportionally with deflection. The reported bending stress is already corrected by the curvature factor q when the load opens the coils; for a closing load that correction is omitted, because coil and inside diameter shrink rather than grow and the standard gives no blanket correction factor for that case. Whether the corrected stress is permissible depends on the actual Rm of the wire used -- this calculator does not automatically compare against a material strength.
Diameter in millimetres, moment in N*mm, moment rate in N*mm/deg, angle in degrees, stress in MPa.
Designing a torsion spring for a known moment-angle ratio, for example in return levers, flaps, hinges, ratchet mechanisms or clamping arms.
DIN EN 13906-3:2014-06, moment-rate, moment and bending-stress equations with curvature correction (notation follows the standard).
Common mistake: Don't neglect winding direction -- a torsion spring is normally rated for one loading direction only; loading it the other way opens or closes the coils differently than intended and can exceed allowable stress or impermissibly change coil diameter. Friction at a mandrel or guide is not captured by the pure characteristic calculation and makes the real required moment higher than the theoretical value. Don't treat the calculated stress as automatically permissible -- it must still be checked against the Rm of the actual wire used (see the spring-wire strength calculator).
From the moment rate RM = E*d^4/(3667*D*n) and the rotation angle alpha via M(alpha) = RM*alpha -- proportional to angle within the linear range of the characteristic.
The slope of the moment-angle curve: the additional moment per degree of rotation. It depends on wire diameter to the fourth power, but only linearly on coil diameter and coil count.
In bending. Despite the name "torsion spring", rotating the legs bends the wire rather than twisting it -- the stress formula sigma = 32M/(pi*d^3) is a bending, not a torsion, equation.
Rotating the legs makes the wire longer and thinner (opening load) or shorter and thicker (closing load), changing coil and inside diameter accordingly. For an opening load, inside diameter shrinks while the coil moves against its guide mandrel.
A torsion spring should only ever be loaded in the direction it was wound for -- usually so the coils open during operation, since that is the case the standard's curvature factor covers. Loading it the other way is not covered normatively and can produce uncontrolled behaviour.
A helical torsion spring (this calculator) stores energy as bending of a coiled wire and is loaded through legs. A torsion bar is a straight rod loaded in pure torsion -- mechanically and analytically a different component.
The calculator describes the linear moment-angle characteristic of the spring body between two working points and accounts for curvature when the coils are opened.
Leg bending, supports, friction, hysteresis, and a complete dynamic verification are outside the declared scope.