Disclaimer: The information on this page has not been checked by an independent person.   Use this information at your own risk.

Click arrows to page adverts

Spring Index

Torsion Spring Design


Springs used to apply torque or store rotational energy are generally called torsion or double torsion springs.   Torque by definition is a force that produces rotation.   A torsion spring exerts a force (torque) in a circular arc, and the arms rotate about the central axis.   The stress is in bending, not in torsion.   It is customary to specify torque with deflection or with the arms at a definite position.  

Torsion bar

The torsion bar is the simplest form or torsion beam.   It comprises a solid or hollow bar which is stressed in torsion within its elastic limit.


P = Force on lever arm (m).
T = Torque resulting from force = PR (Nm)
R = lever radius (m)
D= bar outside diameter (m)
d= bar inside diameter (m)
L = bar length (m)
kL = Linear Spring rate || Force based = P / δ (Nm) ....Torque Based = T / δ (N)
kA = Angular Spring rate || Forced based = P / θ (N)....Torque Based = T / θ (Nm)
G = Modulus of Rigidity (N/mm2)(m)
τ = Allowable shear stress
θ = Deflection (radians)
δ = Linear Deflection= θ. R(m )

Solid Bar
Force BasedTorque Based

Deflection δ = 32PR2 L /( π GD 4 )

Angular Deflection θ = 32PR L /( π GD 4 )

Linear spring rate k L = P / δ = πG D4 /(32R 2 L)

Angular spring rate k A = P / θ =πG D4 /(32RL)

Maximum Load Pmax = π D3 τ /(16 R)

Deflection δ = 32TRL /( π GD 4 )

Angular Deflection θ = 32TL /( π GD 4 )

Linear spring rate k L = T / δ = πG D4 /(32RL)

Angular spring rate k A = T/ θ = πG D4 /(32L)

Maximum Torque Tmax = π D3 τ /16

Hollow Bar
Solid Bar
Force BasedTorque Based

Deflection δ = 32PR2 L /( π G(D 4 - d 4 )

Angular Deflection θ = 32PRL /( π G(D 4 - d 4 )

Linear Spring Rate k L = P / δ= πG (D 4 - d 4) /(32R 2 L)

Angular Spring Rate k A = P / θ = πG (D 4 - d 4) /(32RL)

Maximum Load Pmax = π(D 4 - d 4) τ /(16DR)

Deflection δ = 32TR L /( π G(D 4 - d 4 )

Angular Deflection θ = 32TL /( π G(D 4 - d 4 )

Linear Spring Rate k L = T / δ= πG (D 4 - d 4) /(32RL)

Angular Spring Rate k A = T / θ = πG (D 4 - d 4) /(32L)

Maximum Torque Tmax = π(D 4 - d 4) τ /(16D)

Torsion Spring

A typical torsion helical spring is shown below.  There are a wide variety of coil end configurations to suit different applications and a torsion spring is usually positioned on a shaft.  The coils are usually close wound as are tension springs but they generally do not have any initial tension unlike tension springs.

The primary stress induced in torsion spring is a bending stress in the wire .  This is not the case for the tension and compression helical springs for which the primary stress is a torsional (shear) stress.  During forming residual stresses are built up in the winding process.  These residual stresses are in the same direction but of opposite sign to the working stresses resulting when the spring is loaded causing the coils to tighten.  Torsion springs are stronger as a result and they are often designed to work at, or above the yield strength.


C = Spring Index D/d
d = wire diameter (m)
D = Spring diameter (m)
Di = Spring inside diameter (m)
Dil = Spring inside diameter (loaded ) (m)
E = Young's Modulus (N/m2)
I = Moment of Inertia of wire(m4 )
F = applied Force (N)
G = Modulus of Rigidity (N/m2)(m)
ka = Angular spring rate (stiffness) M /θ (Nm /radian)
L = length (m)
M = Moment (Torque) = RF (Nm)
n = Number of active coils
y = distance from neutral axis to outer fibre of wire (m)
τ = Allowable shear stress (N/m2 )
θ = Deflection (radians)
σ = Bending stress (N/m2 )

Note: metres (m) have been shown as the units of length in all of the variables above for consistency.   In most practical calculations milli-metres will be more convenient.

Torsion Spring Formulae

The spring stress concentration factors Ki =

The maximum bending stress is at the inner fibre of the coil and equals

The angular spring rate ka =

Torsion springs are often used over shafts.  It is important that the spring inside diameter, when fully loaded is no t equal to, or less than the shaft diameter.  If this happens the spring will fail.   The inside diameter of the loaded tension spring is

Links to Spring Design
  1. Engineers Edge- ...Torsion Spring Calculator
  2. Torsion Springs ...Notes on how to make torsion springs
  3. Mitcalc ...Reasonably priced excel based software ( 30 day trial offer)

This Page is being developed

Spring Index

Send Comments to Roy Beardmore

Last Updated 31/05/2011