Torsion Spring Calculator

Enter the wire diameter, mean coil diameter, number of active coils, modulus of elasticity, and angular deflection to compute a helical torsion spring's spring rate, torque, and bending stress.

Quick Facts

Spring rate
k = (E × d⁴) / (10.8 × D × N)
Torque per revolution for a round-wire helical torsion spring (Shigley's design equation).
Torque-deflection law
M = k × θ
Linear within the spring's elastic limit, like a rotational Hooke's law.
Bending stress
σ = Ki × 32M / (πd³)
Ki corrects for the higher stress on the inner fiber caused by coil curvature.
Spring index
C = D / d, ideally 4-14
Too low is hard to coil; too high buckles or tangles.

Your Results

Calculated
Spring Rate (k)
-
Torque per degree of twist
Torque at Deflection (M)
-
M = k × θ
Bending Stress (σ)
-
Curvature-corrected, at the inner fiber
Spring Index (C = D/d)
-
Recommended range: 4-14

Ready

Enter the spring's dimensions and deflection, then press Calculate.

How the Torsion Spring Calculator Works

A helical torsion spring is a coil of wire that resists twisting rather than pushing or pulling — the counterbalance spring in a garage door, the coil in a clothespin, and the wind-up spring in a mousetrap are all torsion springs. When the free leg is rotated by an angle θ, the coil pushes back with a resisting torque M. Within the elastic range this relationship is linear, so the spring behaves like a rotational version of Hooke's law: M = k × θ, where k is the spring rate. This calculator uses the standard round-wire torsion spring design equations (from Shigley's Mechanical Engineering Design) to compute the spring rate, the torque produced by a given deflection, and the resulting bending stress in the wire.

Spring rate and torque from wire and coil geometry

The spring rate of a round-wire helical torsion spring, expressed as torque per revolution, is k = (E × d⁴) / (10.8 × D × N), where E is the modulus of elasticity of the wire material, d is the wire diameter, D is the mean coil diameter, and N is the number of active coils. The wire diameter has the strongest effect on stiffness because it enters to the fourth power — doubling d makes the spring roughly 16 times stiffer — while a larger coil diameter or more active coils both make the spring softer. Once k is known, the torque produced by any angular deflection follows directly: M = k × (θ / 360°) when θ is entered in degrees, or M = k × (θ / 2π) when θ is entered in radians.

Checking the bending stress

Torsion springs fail by bending overstress at the inner fiber of the coil, not by the shear stress that governs compression and extension springs. The nominal bending stress, 32M / (πd³), is multiplied by a curvature correction factor Ki = (4C² − C − 1) / (4C(C − 1)), where C = D/d is the spring index, because the coil's curvature concentrates extra stress on the inside of each turn. Compare the corrected stress to the wire material's allowable bending stress — commonly around 50-65% of its ultimate tensile strength for static loading — and keep the spring index between roughly 4 and 14 so the spring can be coiled and manufactured reliably.

Frequently Asked Questions

What is a torsion spring and how is torque related to angular deflection?
A torsion spring is a helical coil of wire that resists twisting and stores energy as its free leg is rotated. Within the elastic range the applied torque M is directly proportional to the angular deflection θ: M = k × θ, where k is the spring rate (torque per unit of rotation). The spring is loaded by twisting its legs, not by axial push or pull.
What is the formula for torsion spring rate?
For a round-wire helical torsion spring, the spring rate per revolution is k = (E × d⁴) / (10.8 × D × N), where E is the wire material's modulus of elasticity, d is the wire diameter, D is the mean coil diameter, and N is the number of active coils. The 10.8 constant (versus the theoretical 10.19) accounts for friction between the coils and the mandrel/arms, per Shigley's standard torsion spring design equations.
Why does bending stress need a curvature correction factor (Ki)?
Because the wire is curved into a coil, stress is not distributed evenly across its cross-section the way it is in a straight beam — the inner fiber of each turn sees higher stress than the outer fiber. The correction factor Ki = (4C² − C − 1) / (4C(C − 1)), where C = D/d is the spring index, is multiplied into the nominal bending stress formula 32M/(πd³) to account for this concentration at the critical inner fiber.
What spring index (C = D/d) should I use?
Most helical torsion springs use a spring index between about 4 and 14. Below roughly 4 the wire becomes difficult and expensive to coil; above roughly 14 the spring tends to buckle sideways, tangle, and lose dimensional consistency.

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