How this instrument works
A torsion spring resists rotation the way an ordinary coil spring resists stretching, except the load is a moment and the response is an angle rather than a length. Its stiffness, kθ, is rated in newton-metres per radian — the torque needed to wind the spring through one full radian of twist — and inside the working range that relationship stays a straight line: T = kθ·θ. Double the twist and the spring pushes back twice as hard. This instrument reads out that push-back, plus the energy already banked while getting there, PE = ½kθθ² — the rotational counterpart of a coil compressed along its own axis, reading force through angle instead of through straight-line displacement.
Despite the name, the wire inside a torsion spring is not really working in torsion. Apply torque to its legs and the coils tighten or loosen about their own axis, which puts the wire's cross-section in bending rather than shear — the same stress state you'd get flexing a straight rod, coiled into a helix and stacked in series. That is why catalogue formulas for kθ carry the material's modulus of elasticity, E, rather than the shear modulus, G, that governs a twisted shaft or a torsion bar, and why wire diameter enters to the fourth power while coil diameter and active coil count sit underneath, in the denominator.
Two limits sit under that straight line. Good practice loads a torsion spring so the coils wind tighter, shrinking their diameter slightly as the legs close — coiling leaves a favourable residual stress on the wire's inner surface in that direction, and manufacturers specify it as a wind sense, left-hand or right-hand, for exactly this reason. Load it backwards, so the coils open, and that residual stress works against you instead. Second, the legs need a mandrel or guide to stop the coil bowing sideways under load, and once deflection runs much past roughly 180° per active coil, or the wire yields, the torque-angle line bends and kθ stops being one constant.
- Enter your spring's rating into Torsional spring constant, N·m/rad — taken from a datasheet, or measured by winding a known torque onto the spring and reading the angle it turns through.
- Enter how far the spring is wound into Twist angle, measured from its free, unloaded position. The default unit is degrees; switch to radians if that's how your source data is given.
- Read Restoring torque — the push-back the spring delivers at that exact angle.
- Read Stored elastic energy in joules, the work already invested in winding the spring out to that angle.
Worked example — a 50 N·m/rad garage-door spring at 30°
Take a torsion spring rated at 50 N·m/rad — a stiffness in the range of a residential garage-door spring — partway through its travel at 30° of twist, or 0.523598775598299 rad. Enter 50 into Torsional spring constant, N·m/rad and 30 into Twist angle. The formula turns that into T = 50 × 0.523598775598299 = 26.1799387799 N·m of restoring torque, the push-back the spring delivers at that exact point in its wind, which is what a properly balanced door relies on to counter its own weight.
Stored energy follows the same two inputs through PE = ½ × 50 × 0.523598775598299² = 6.8538919452 J. That energy did not appear all at once; it accumulated across the whole twist from zero, growing with the square of the angle rather than in step with it, so the last few degrees of any wind bank noticeably more joules than the first few did. A technician winding this spring by hand feels that curve directly — the final turn always takes visibly more effort than the first.
Questions
Why is the spring rating in newton-metres per radian rather than per degree?
Because the formula's derivation runs in radians — a pure ratio of arc length to radius with no unit of its own — so T = kθ·θ only returns the correct torque when θ is fed in as radians. Manufacturers often publish rate per degree, or per full turn, instead: multiply a per-degree figure by 57.2958, or a per-turn figure by roughly 6.2832, before it goes into this field.
Is a torsion spring's wire actually being twisted?
Not in the way the name suggests. Applying torque to the legs winds the coils tighter or looser about their own axis, which loads the wire's cross-section in bending rather than in torsional shear — the same stress state you'd get flexing a straight rod, coiled into a helix. That is why the material property behind a real spring-rate calculation is the modulus of elasticity, E, not the shear modulus, G, that governs a twisted shaft.
Which direction should a torsion spring be loaded?
Whichever direction winds the coils tighter, shrinking their diameter slightly as the legs close. Coiling leaves a favourable residual stress on the wire's inner surface in that direction; loading it the other way, so the coils open and the diameter grows, works against that residual stress and shortens fatigue life. Datasheets specify a wind sense, left-hand or right-hand, for exactly this reason, and it is not interchangeable.
What happens if I twist the spring too far?
Past a certain point the torque-angle line stops being straight. Most catalogue torsion springs are rated for a maximum deflection of roughly 180° per active coil; push beyond that, or past the wire's yield stress, and the spring takes a permanent set, returning to a new, more open rest position rather than its original one. This formula only holds inside that linear, fully elastic range.
How do I find kθ for a spring with no datasheet?
Measure it directly: clamp one leg, apply a known torque to the other with a torque wrench or a calibrated arm and weight, and read the angle it turns through. Dividing that torque by that angle gives kθ in N·m/rad, the same empirical route used to characterise any unmarked spring. Take the reading well inside the travel you actually intend to use.
Does PE = ½kθθ² tell me the force released if the spring breaks?
It gives the theoretical maximum, everything currently stored at that angle, not what actually reaches you. Friction, wire fracture and the flying mass of the coil itself absorb part of it as heat and as kinetic energy of fragments. Treat the figure as an upper bound, which is exactly why garage-door torsion springs are installed with a restraining cable run through their centre.