Torsion springs and torsion bars are used to store and apply rotational energy through torque. This page provides engineering equations, stress relationships and design considerations for torsion spring systems used in mechanical engineering applications.
A helical torsion spring stores energy as its coils are angularly deflected. Despite its name, the principal stress in the wire of a helical torsion spring is bending stress. A torsion bar, by contrast, is a straight solid or hollow bar whose material is stressed in torsional shear.
Springs used to apply torque or store rotational energy are commonly referred to as torsion springs. Single-body and double-torsion arrangements are used depending on the application.
A helical torsion spring exerts a torque as its arms rotate about the spring axis. The applied torque causes bending of the coiled wire rather than the torsional shear that is the principal stress in conventional compression and extension spring wire.
Torsion springs are therefore normally specified in terms of the required torque at one or more angular positions, together with the spring geometry, direction of winding, leg arrangement and available operating angle.
A torsion bar is one of the simplest forms of rotational spring. It consists of a solid or hollow bar subjected to torque within its elastic working range. One end is normally restrained while torque is applied to the other end, either directly or through a lever arm.
| Symbol | Description |
|---|---|
| P | Force applied to the lever arm (N) |
| T | Applied torque = PR (N m) |
| R | Lever-arm radius (m) |
| D | Outside diameter of bar (m) |
| d | Inside diameter of hollow bar (m) |
| L | Effective torsion-bar length (m) |
| G | Shear modulus / modulus of rigidity (N/m2) |
| J | Polar second moment of area (m4) |
| τ | Maximum shear stress (N/m2) |
| θ | Angular deflection (radians) |
| δ | Linear displacement at lever radius R (m) |
| kθ | Torque-based angular spring rate = T/θ (N m/rad) |
| kL | Force-based linear spring rate = P/δ (N/m) |
For small angular deflections, the displacement at the end of the lever is:
δ ≈ Rθ
The fundamental elastic torsion relationship is:
θ = TL / (GJ)
For a solid circular bar:
J = πD4 / 32
and therefore:
θ = 32TL / (πGD4)
If the torque is produced by a force P acting at radius R:
T = PR
θ = 32PRL / (πGD4)
For small rotations, the corresponding tangential displacement at radius R is:
δ = 32PR2L / (πGD4)
The torque-based angular spring rate is:
kθ = T / θ = πGD4 / (32L)
The force-based angular rate at a lever arm of radius R is:
P / θ = πGD4 / (32RL)
The force-based linear spring rate at the end of the lever is:
kL = P / δ = πGD4 / (32R2L)
The maximum shear stress occurs at the outside surface:
τmax = 16T / (πD3)
For an allowable shear stress τallow, the corresponding maximum elastic torque is:
Tmax = πD3τallow / 16
and, for a lever arm of radius R:
Pmax = πD3τallow / (16R)
For a hollow circular bar having outside diameter D and inside diameter d:
J = π(D4 - d4) / 32
The angular deflection is:
θ = 32TL / [πG(D4 - d4)]
or, for T = PR:
θ = 32PRL / [πG(D4 - d4)]
For small angular deflections, the tangential displacement at the lever end is:
δ = 32PR2L / [πG(D4 - d4)]
The torque-based angular spring rate is:
kθ = T / θ = πG(D4 - d4) / (32L)
The force-based angular rate at lever radius R is:
P / θ = πG(D4 - d4) / (32RL)
The force-based linear spring rate is:
kL = P / δ = πG(D4 - d4) / (32R2L)
The maximum shear stress occurs at the outside surface:
τmax = 16TD / [π(D4 - d4)]
For an allowable shear stress τallow:
Tmax = π(D4 - d4) τallow / (16D)
and:
Pmax = π(D4 - d4) τallow / (16DR)
A helical torsion spring consists of a coil with arms or legs that apply and react a torque. Many different leg and end configurations are possible to suit the surrounding mechanism, and the spring is commonly mounted over a shaft, arbor or other supporting feature.
The body coils are normally close wound. Unlike many close-wound extension springs, however, a torsion spring does not normally rely on initial axial tension between its coils.
Despite the name torsion spring, the principal stress produced in the coiled wire by the operating torque is a bending stress. This differs from a helical compression or extension spring, where the principal working stress in the wire is torsional shear.
The forming process can also produce residual stresses in the wire. When a torsion spring is loaded in its preferred direction so that the coil winds tighter, these residual stresses can be favourable to its load-carrying behaviour.
| Symbol | Description |
|---|---|
| C | Spring index = D/d |
| d | Wire diameter |
| D | Mean coil diameter |
| Di | Inside coil diameter in the free condition |
| Dil | Inside coil diameter under load |
| E | Young's modulus |
| F | Applied force |
| I | Second moment of area of the wire section |
| Ki | Inner-fibre curvature/stress correction factor |
| M | Applied moment / torque = RF |
| n | Number of active body coils |
| R | Effective moment arm of the applied force |
| θ | Angular deflection |
| σ | Bending stress in the spring wire |
| ka | Angular spring rate = M/θ |
For practical spring calculations, N, mm and N/mm2 are convenient units. With this system, torque is expressed in N mm.
For round-wire cylindrical helical torsion springs:
C = D / d
Curvature increases the bending stress at the inside surface of the coil. For round wire, the inner-fibre stress correction factor is:
Ki = (4C2 - C - 1) / [4C(C - 1)]
The corresponding outer-fibre correction factor is:
Ko = (4C2 + C - 1) / [4C(C + 1)]
For the usual loading direction, the inside fibre is generally the more highly stressed surface.
The nominal bending stress in round wire subjected to moment M is:
σ = 32M / (πd3)
Including the inner-fibre curvature correction:
σmax = Ki 32M / (πd3)
For a simplified round-wire helical torsion spring in which the flexibility of the legs is neglected, the angular deflection of the body coils may be approximated by:
θ = 64MDn / (Ed4)
where θ is in radians when consistent units are used.
Since:
ka = M / θ
the corresponding simplified angular spring rate is:
ka = Ed4 / (64Dn)
This is a simplified body-coil relationship. In a practical torsion spring, the spring arms can also deflect and therefore contribute to the total compliance of the spring.
A torsion spring is frequently mounted over a shaft or arbor. When the spring is loaded in the direction that winds the coils tighter, the coil diameter decreases.
The approximate decrease in mean coil diameter for an angular movement θ in radians can be represented by:
ΔD = θD / (2πn)
The approximate loaded mean diameter is therefore:
DL = D - ΔD
and the approximate loaded inside diameter is:
Dil = DL - d
Combining these relationships:
Dil = D - d - θD/(2πn)
The simple equations above are useful for preliminary design and checking, but a practical torsion spring also requires consideration of:
For cyclic applications see Fatigue in Springs, and for material properties see Spring Materials.
The principal European standard for calculation and design of cylindrical helical torsion springs made from round wire and bar is:
EN 13906-3 covers the calculation and design of cold- and hot-coiled cylindrical helical torsion springs having a linear characteristic and made from round wire or bar of constant diameter.