Torsion Spring
Compute the bending stress σ, angular deflection θ and torsion rate of a round-wire torsion spring under moment M, using standard torsion-spring formulas with the Wahl stress-correction factor K.
Inputs
Formula
c=D/d; K=(4c−1)/(4c−4); σ=K·32M/(πd³); θ=32MDN/(Ed⁴); rate=M/θ
Fundamentals
Torsion springs store energy in a wind-up angle about the axis — common in hinges, clamps and clocks.
Torque is proportional to angle $M=k_\theta\theta$, $k_\theta$ the torsional stiffness (wire dia, coils, material); it returns on release.
History
Torsion springs powered mainsprings (14th-c. clocks); coiled torsion springs spread with 19th-c. precision making.
Engineering applications
Used in hinges, clips, clutches and timing or return mechanisms.
Glossary
| Torsional stiff $k_\theta$ | Torque per unit angle. |
| Wind angle $\theta$ | Relative rotation of the ends. |
| Restoring torque | Moment returning the mechanism on release. |
How to use
- Fill in Moment M, Mean dia. D, Wire dia. d, Active coils N, Young's mod. E in the Inputs section (watch the unit on each field).
- Click Calculate; the tool evaluates the formula shown above.
- Read Bending stress σ, Deflection θ, Torsion rate in the results area.
Formula notesTorsion spring rate k = E·d⁴/(32·D·n); torque M = k·θ.
Formula · Worked Example · Knowledge
Formula
Round-wire torsion spring: index $c=D/d$, stress factor $K=(4c-1)/(4c-4)$, bending stress $\sigma=K\,32M/(\pi d^3)$, angle $\theta=32MDN/(E d^4)$.
Worked Example
M=5 N·m, D=15, d=1.5, N=5, E=200 GPa → σ≈249 MPa, θ≈97°.
Key Points
- Torsion springs see bending stress; use spring steel.
- Longer legs → larger angle, lower rate.
- Pre-rotate at fit to remove slack.
Parameters
Inputs
| Parameter | Symbol | Unit | Default |
|---|---|---|---|
| Moment M (N·m) | M | N·m | 5 |
| Mean dia. D (mm) | D | mm | 15 |
| Wire dia. d (mm) | d | mm | 1.5 |
| Active coils N | N | 5 | |
| Young's mod. E (GPa) | E | GPa | 200 |
Outputs
| Result | Symbol | Unit |
|---|---|---|
| Bending stress σ | sigma | MPa |
| Deflection θ | theta | ° |
| Torsion rate | rate | N·mm/rad |
Applications
- Common engineering use cases
FAQ
What formula does this tool use?
This tool computes per ISO / AGMA / ASME standard formulas: c=D/d; K=(4c−1)/(4c−4); σ=K·32M/(πd³); θ=32MDN/(Ed⁴); rate=M/θ
How accurate are the results?
Results match input precision, based on SI units and common engineering approximations; for critical duty re-check with a safety factor.
Where is it used?
Common engineering use cases