Torsion Spring Calculator
Enter the wire diameter, outer coil diameter, number of active coils, deflection angle, and wire material to get the spring rate, delivered torque, inner-fiber bending stress, and spring index. The calculator uses the Shigley / SMI beam-bending model with the inner-surface stress correction factor. Switch between metric (mm, N-mm) and imperial (in, lb-in) units and results update instantly.
How torsion springs work
A helical torsion spring stores energy by bending its wire as the coil is twisted. Unlike compression or extension springs, which load the wire in shear, a torsion spring loads the wire in bending, so the key property is the modulus of elasticity (Young's modulus, E) rather than the shear modulus. When a moment is applied to one arm while the other is fixed, each active coil bends slightly, and the cumulative deflection is proportional to the load. The relationship is linear for small deflections, so the spring can be fully characterized by a single spring rate: torque per degree of rotation.
How to use this calculator
Enter the wire diameter (d), the outer coil diameter (OD), the number of active coils (Na), and the expected deflection angle. Choose the wire material so the correct modulus of elasticity is applied. The calculator instantly returns the spring rate, the torque at that angle, the inner-fiber bending stress (with the Wahl stress-correction factor Ki), and the spring index C = D/d. The stress-to-tensile-strength ratio tells you whether the design is safe for static or high-cycle use. Switch between imperial (inches, lb-in, psi) and metric (mm, N-mm, MPa) units and all results convert automatically.
Spring rate formula and stress correction
The spring rate for a helical torsion spring is k = E*d^4 / (10.8 * D * Na), where D = OD - d is the mean coil diameter and the factor 10.8 = pi^2 / (32/degrees-radian-conversion) comes from the derivation for radians converted to degrees. The inner-surface stress correction factor Ki = (4C^2 - C - 1) / (4C*(C-1)) accounts for the curvature effect that concentrates stress on the inside of the coil. Maximum bending stress is sigma = Ki * 32M / (pi * d^3). For static applications the stress should stay below about 80% of the ultimate tensile strength. For high-cycle fatigue applications, aim for 35 to 45% to achieve infinite life.
Spring index and manufacturability
The spring index C = D/d is the ratio of mean coil diameter to wire diameter. It controls both stress concentration and ease of manufacture. A spring index below 4 indicates very tightly wound coils that are difficult to form accurately and prone to cracking during winding. An index above 16 produces coils that are hard to maintain in shape and tangle during manufacturing. The ideal range for most production springs is 4 to 16, with 6 to 12 being the most common. If your design falls outside these limits, adjusting wire diameter is usually the simplest fix. Also ensure at least 10% radial clearance between the inner diameter of the spring and the shaft it rides on, because the coil diameter decreases as the spring winds up under load.
Typical wire material properties
| Material | E (psi) | E (MPa) | Approx. Su (ksi) | Best for |
|---|---|---|---|---|
| Music Wire | 11,500,000 | 79,290 | 190 | General high-precision springs |
| Oil-Tempered Wire | 10,000,000 | 68,950 | 170 | General industrial springs |
| Stainless 302/304 | 11,200,000 | 77,220 | 170 | Corrosion-resistant applications |
| Chrome Silicon | 9,500,000 | 65,500 | 200 | High temperature / high stress |
| Phosphor Bronze | 6,500,000 | 44,820 | 120 | Electrical, non-magnetic, corrosion-resistant |
Young's modulus (E) and approximate tensile strength for common torsion spring alloys.
Frequently asked questions
What is a torsion spring?
A torsion spring is a helically coiled wire spring that exerts a torque (rotational force) when its arms are deflected. The wire is loaded in bending rather than in torsion despite the name. Common uses include clothespins, mousetraps, door hinges, garage door counterbalances, and automotive suspension components. The spring returns the arms to their free position when the load is released.
What does spring rate mean for a torsion spring?
The spring rate (k) is the torque required to deflect the spring by one degree. If a spring has a rate of 0.1 lb-in/deg, deflecting it 90 degrees delivers 9 lb-in of torque. The rate is set by the wire material, wire diameter, coil diameter, and number of active coils. Increasing wire diameter or decreasing coil diameter or active coil count all increase the rate.
What is the spring index and why does it matter?
The spring index C = D/d is the ratio of mean coil diameter to wire diameter. It affects the stress concentration inside the coil and how easy the spring is to manufacture. Values below 4 cause excessive stress and cracking during winding. Values above 16 make coils hard to hold to tolerance and prone to tangling. Most torsion springs are designed with an index between 4 and 16.
How does deflection direction affect a torsion spring?
Torsion springs perform best when deflected in the winding direction (closing the coil). This reduces the coil diameter as it winds up, which is normal behavior. Deflecting in the unwinding direction opens the coil, increases the diameter, and builds stress differently. If the application requires counter-winding direction, reduce the maximum load by roughly 20% as a safety margin, or specify left-hand wound springs if the application allows it.
What stress level is safe for a torsion spring?
For static or rarely cycled applications, the inner-fiber bending stress should stay below about 80% of the ultimate tensile strength of the wire. For springs that cycle frequently (millions of cycles), the stress should be held to 35 to 45% of tensile strength to achieve practical fatigue life. The approximate tensile strength depends on alloy and wire diameter; music wire and chrome silicon offer the highest strength, while phosphor bronze is significantly lower.
Why does the coil diameter change under load?
When a torsion spring is deflected in the winding direction, each coil bends tighter, decreasing the overall coil diameter. The change in mean diameter is approximately D' = (Nb * D) / (Nb + theta_coils), where Nb is the number of body coils and theta_coils is the deflection expressed in full turns. This matters for shaft fit: the inner diameter at maximum deflection must still be larger than the shaft, so designers typically leave 10% radial clearance in the free state.
Which wire material should I choose?
Music wire (ASTM A228) offers the highest tensile strength per diameter and is the default choice for precision, general-purpose, and high-cycle springs up to about 120 deg C. Oil-tempered wire is a cost-effective alternative for static or low-cycle industrial applications. Stainless steel 302/304 resists corrosion and is preferred in food, medical, and outdoor environments. Chrome silicon handles elevated temperatures up to about 250 deg C and is used in high-temperature or very high-stress applications. Phosphor bronze is chosen when electrical conductivity, non-magnetic properties, or marine corrosion resistance are required.