Torsional rigidity: Definition, Formula, Units, Examples [with Pdf]

What is Torsional rigidity?

Torsional rigidity is the product of shear modulus (G) and polar moment of inertia (J). The torsional rigidity shows the resistance offered by a material to angular deformation.

Torsional rigidity is also defined as the torque required to produce a unit radian angle of twist per unit length of the shaft.

The term torsional rigidity is expressed as,

Torsional rigidity = G x J

As GJ = `\frac{TL}{\theta }` ——– (∵ Torsional equation `\frac{T}{J}` = `\frac{G\theta }{L}`)

Thus the equation of torsional rigidity can also be written as,

Torsional rigidity = `\mathbf{\frac{TL}{\theta }}`

Where,
L = Length of shaft (mm)
θ =  Angle of twist (Radian)
T = Torque (N.m)

Torsional rigidity equation:

The torsional rigidity of component can be calculated by using the following formula:-

Torsional rigidity = G x J

For the component subjected to the torque of T, the torsional rigidity can be given by,

Torsional rigidity = `\mathbf{\frac{TL}{\theta }}`

Torsional rigidity units:

SI unit:-

In the SI system, the unit of shear modulus (G) is N/m² or Pascal and the unit of polar moment of inertia is m⁴. Thus the unit of torsional rigidity becomes,

GJ = `\frac{N}{m^{2}}`.m⁴ = N.m²

Hence the SI unit of torsional rigidity is N.m².

FPS unit:-

In the FPS system, the unit of shear modulus is lb/ft² and the unit of polar moment of inertia is ft⁴.

Thus the unit of torsional rigidity becomes,

GJ = `\frac{lb}{ft^{2}}.ft^{4}` = lb.ft²

Hence the FPS unit of torsional rigidity is lb.ft².

Torsional rigidity dimensional formula:

The dimensional formula of shear modulus is [ML⁻¹T⁻²] and the dimensional formula of polar moment of inertia is [M⁰L⁴T⁰].

Therefore the dimensional formula of modulus of rigidity if given by,

Torsional rigidity = GJ = [ML⁻¹T⁻²][M⁰L⁴T⁰]

= [ML³T⁻²]

Therefore the dimensional formula of the torsional rigidity is [ML³T⁻²].

Torsional rigidity examples:

1] A circular shaft of radius 36 mm is made of aluminum with a shear modulus of 69 Gpa. Find the torsional rigidity of the shaft.

Given:
d = 36 mm
G = 69 Gpa = 69 x 10³ N/mm²

Solution:-

The polar moment of inertia for the shaft is given by,

J = `\frac{\pi }{32}d^{4}` = `\frac{\pi }{32}36^{4}` = 164895.9 mm⁴

The torsional rigidity of the shaft is,

Torsional rigidity = GJ = (69 x 10³) x (164895.9)

= 11.377 x 10⁹ N.mm²

Torsional rigidity = 11377 N.m²

2] Find torsional rigidity by using following data,
T = 80 N.m
θ = 1°
L = 1 m

Given:
T = 80 N.m
θ = 1° = (1 x `\frac{\pi }{180}`) radian = 0.017 radian
L = 1 m

Solution:-

The torsional rigidity can be calculated as,

Torsional rigidity = `\frac{TL}{\theta }` = `\frac{800 \times 1}{0.017}`

= 47058.82 N.m²

Torsional rigidity = 47058.82 N.m²

FAQ’s:

  1. What do you mean by torsional rigidity?

    Torsional rigidity is the resistance offered by an object to angular deformation which is equivalent to the product of shear modulus and polar moment of inertia.

  2. What is unit of torsional rigidity?

    The SI unit of torsional rigidity is N.m² and the FPS unit is lb.ft².

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