Calculate the spring rate (stiffness) for helical torsion springs based on material and geometric properties.
Last updated: March 2026 | By Summacalculator
Steel: ~81,000 MPa, Stainless: ~77,000 MPa
A torsion spring is a helical spring designed to resist twisting (angular) loads. Unlike compression and tension springs that resist linear forces, torsion springs generate a restoring torque proportional to the angle of twist applied to them. They are manufactured by tightly winding wire around a cylindrical mandrel to create a helix, then removing the mandrel to leave a precise coil geometry. The spring rate (stiffness) determines how much torque is required to twist the spring by one radian. Torsion springs are commonly used in mechanical devices like clothespins, mouse traps, door hinges, and automotive suspension bars where a rotational restoring force is needed.
The spring rate depends on four key parameters: the wire diameter (d), which controls the wire's bending stiffness; the mean coil diameter (D), which determines the lever arm for twisting; the number of active coils (n), where more coils reduce stiffness; and the shear modulus (G), a material property describing resistance to twisting deformation. The formula k = (G × d⁴) / (64 × D × n) shows that spring rate increases dramatically with wire diameter (fourth-power dependence) but decreases linearly with coil diameter and coil count. Note: torsion spring stiffness depends on shear modulus G (the resistance to shear stress), not Young's modulus E (which applies to linear compression/tension). For steel, G ≈ 81,000 MPa compared to E ≈ 200,000 MPa. Engineers select torsion spring parameters based on required stiffness, available space, and material availability. High-performance springs use materials like chrome vanadium or stainless steel for improved fatigue resistance under cyclic loading.
Step 1: Enter the wire diameter in millimeters. This is the diameter of the individual wire forming the spring coils. For precision springs, this is typically 0.5 – 5 mm depending on application requirements.
Step 2: Enter the mean coil diameter (centerline of the wire to centerline) in millimeters. This is the effective diameter around which the wire is wound. For a spring with 20 mm outer diameter and 2 mm wire, the mean diameter is approximately 16 mm.
Step 3: Enter the number of active coils. This is the count of coils that contribute to the spring's flexibility, excluding any end coils that are ground or not fully active. Typical values range from 5 to 15 coils.
Step 4: Enter the shear modulus (G) in MPa. For steel, use 81,000 MPa; for stainless steel, 77,000 MPa. The shear modulus describes the material's resistance to twisting deformation—use this value, not Young's modulus. The calculator displays the spring rate in N⋅mm/radian: the torque (in N⋅mm) required to twist the spring by 1 radian.
A designer is creating a spring-loaded hinge for an oven door that requires a restoring torque of 5 N⋅m for a 90-degree (1.57 radian) twist. To meet this requirement with a standard steel wire spring, calculate the required spring rate, then use this calculator to verify a candidate design with 2 mm wire diameter, 20 mm mean coil diameter, and 8 active coils.
Torsion springs resist rotational (twisting) forces and store energy by twisting around their axis, while compression springs resist linear forces applied along their axis. Torsion springs typically have legs or arms for loading, whereas compression springs are loaded end-to-end. Torsion springs are ideal for applications requiring angular restoring forces, like hinges or throttle returns.
Wire diameter has a fourth-power effect on spring rate: doubling the wire diameter increases stiffness by 2⁴ = 16 times. This is the most powerful way to adjust spring rate. A 2 mm wire spring is 16 times stiffer than a 1 mm wire spring (all else equal). Wire diameter also affects fatigue resistance and load capacity.
Mean coil diameter (D) is measured from the centerline of the wire on one side of the coil to the centerline of the wire on the opposite side—effectively the diameter of the spring coil path. For a spring with 2 mm wire wound to an outer diameter of 22 mm, the mean diameter is approximately 18 mm (22 mm minus 4 mm for two wire radii).
Active coils are coils that deflect and contribute to spring flexibility. Total coils minus non-deflecting end coils (typically 1-2 end coils are ground flat and do not bend as freely) equals active coils. For a spring with 10 total coils and 2 end coils, active coils = 8. Manufacturers information sheets specify active coils explicitly.
Steel is the most common choice: low-carbon steel for light applications, high-carbon steel for standard use, and stainless or chrome-vanadium for fatigue-critical and corrosive environments. Titanium offers superior strength-to-weight but at higher cost. Material choice affects material stiffness (E): steel approx 200 GPa, stainless approx 190 GPa, titanium approx 103 GPa.
Torsion springs can typically withstand 1,000,000 to 10,000,000 cycles of loading in normal applications, depending on material grade, wire quality, and stress levels. Life is governed by fatigue strength—the stress amplitude, mean stress, and surface finish all affect endurance limit. Design for less than 50% of material yield strength to maximize cycle life.
No—more coils actually decrease spring rate (make the spring softer). Spring rate is inversely proportional to coil count: doubling coils halves stiffness. Longer springs (more coils) distribute stress over more material, reducing local bending stress but also reducing overall stiffness. To increase stiffness, use thicker wire or smaller coil diameter.
Young's modulus (E) describes a material's resistance to linear stress (compression or tension), while shear modulus (G) describes resistance to shear stress (twisting or sliding). For torsion springs, the relevant property is shear modulus G, not Young's modulus E. Typical values: steel E ≈ 200,000 MPa but G ≈ 81,000 MPa. Using the wrong modulus (E instead of G) would overestimate spring stiffness by a factor of ~2.5 and produce incorrect results.