A helical torsion spring produces a turning moment when its legs rotate. Although the spring is called a torsion spring, the wire in the coil body is principally bent; it is not analyzed like a straight torsion bar. This reference summarizes several equations published by the Spring Manufacturers Institute (SMI), with symbols and unit conventions kept explicit. These relationships are useful for preliminary sizing and checks, not a complete fatigue-rated spring design.

Torsion spring design formulas

Spring rate and angular units

SMI gives a helical torsion spring rate approximation k = E d⁴ / (10.8 Na D). E is Young’s modulus, d is wire diameter, Na is the effective active-coil count, and D is mean coil diameter. With E in N/mm² and all lengths in millimetres, the rate has units of N·mm per turn. The source’s factor 10.8 is larger than its theoretical factor 10.2 to allow for friction between adjacent coils and between the spring body and the support shaft.

The unit “per turn” matters. If a calculation is expressed per degree, divide the per-turn value by 360; if it is per radian, use 2π radians per turn. Do not copy a rate into a drawing or data sheet without stating whether the angle is in turns, degrees, or radians. Torque at a deflection is M = k Θ when Θ uses the same angular basis as k and is measured from the spring’s free position.

Worked rate example

For a hypothetical round-wire spring with E = 206,000 N/mm², d = 2 mm, D = 20 mm, and Na = 8, the equation gives k = 206,000 × 2⁴ / (10.8 × 8 × 20), or about 1,907 N·mm per turn. That equals about 1.91 N·m per turn, or 5.30 N·mm per degree. These are calculated values for the stated assumed dimensions and modulus only; no spring material condition or allowable stress is certified by the example.

Legs, stress, and energy

The spring legs can bend and add compliance. SMI estimates their contribution as Ne = (L1 + L2)/(3πD), then sets Na = Nb + Ne, where Nb is the number of body coils and L1 and L2 are the moment-arm lengths. If the legs are long, flexible, or loaded differently from the cited arrangement, this simplified contribution may not capture the real behavior. The free-end angle and winding direction are also essential parts of the specification.

Wire stress in the coil body is bending stress. SMI presents a corrected form S = KBID × 32M/(πd³), where KBID is a stress correction factor. The correction factor and sign convention must match the design method. The published article directs readers to SMI’s spring-design reference for the detailed factor; do not silently set it to one or confuse this equation with torsional shear stress in a straight bar.

SMI also gives stored energy capacity per unit volume for round wire as ESC = S²/(8E), omitting the stress concentration factor. This is an energy-density relationship, not total spring energy. A total requires the active material volume and the full geometry and loading path. The square-wire relation has a different coefficient and should not be substituted into a round-wire calculation.

Common design mistakes and limits

Do not use compression-spring equations based on shear modulus and axial force for a helical torsion spring. Do not call the body-wire stress “torsional stress” simply because of the spring name. Do not ignore leg bending, changing inside diameter as the coil winds, support-shaft contact, or interference between adjacent coils. Those effects can change rate, stress, and available angular travel.

Material, heat treatment, wire diameter tolerance, residual stress, surface condition, operating temperature, cycles, dwell time, and corrosion all affect performance. The equations do not provide an allowable stress, fatigue life, relaxation guarantee, or production tolerance. Check the actual spring dimensions and operating envelope with a qualified spring manufacturer, especially for safety-related or high-cycle applications.

Sources

The formulas and notes above follow the Spring Manufacturers Institute’s Winter 2025 torsion-spring article. A separate spring-manufacturer explanation of rate calculations reinforces the need to distinguish linear rate from torque per angular deflection. Confirm symbols, units, and design factors against the method selected for the actual spring.