# Torsional stiffness: Definition, Equation, Units, of shaft with Pdf

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## What is torsional stiffness?

Torsional stiffness is defined as the amount of torque required for twisting an object by unit radian.

It is also known as the ratio of torque to the angular twist.

Hence the torsional stiffness is given by,

Torsional stiffness = \frac{T}{\theta }

## Torsional stiffness formula:

From the definition, the torsional stiffness equation is written as,

Torsional stiffness = \frac{T}{\theta }

From the torsional equation,

\frac{T}{\theta }= \frac{GJ}{L}

Where,
G = Modulus of rigidity
J = Polar moment of inertia
L = Length of shaft

Therefore torsional stiffness equation can be written as,

Torsional stiffness = \frac{T}{\theta }= \frac{GJ}{L}

Hence it is also known as torsional rigidity per unit length of the object.

## Torsional stiffness units:

The SI and FPS units of the Torsional stiffness are as follows:-

i] SI unit:

In SI system the unit of torque is N.m and the unit of the angle of twist is radian, therefore the unit of the torsional stiffness is given by,

Torsional stiffness = \frac{T}{\theta } = N.m/radian

Therefore SI unit of torsional stiffness is N.m/radian.

ii] FPS unit:

In the FPS system, the unit of torque is lb.ft and the unit of the angle of twist is the radian. Hence the unit of the torsional stiffness in the FPS system is,

Torsional stiffness = \frac{T}{\theta } = lb.ft/ radian

Therefore FPS unit of torsional stiffness is lb.ft/radian.

## Torsional stiffness of shaft:

1) Torsional stiffness of solid shaft:

For solid circular shaft, J= \frac{\pi }{32}\times d^{4}

Therefore torsional stiffness of the solid shaft is,

Torsional stiffness = \frac{GJ}{L} = \frac{G}{L}\times [\frac{\pi }{32}\times d^{4}]

Torsional stiffness = \frac{G\pi d^{4}}{32L}

2) Torsional stiffness of hollow circular shaft:

For hollow circular shaft with an outside diameter of do and inside diameter of di,

J = \frac{\pi }{32}\times (do^{4}-di^{4})

Hence torsional stiffness of the hollow shaft is,

Torsional stiffness = \frac{GJ}{L} = \frac{G}{L}\times [\frac{\pi }{32}\times (do^{4}-di^{4})]

Torsional stiffness = \frac{G\pi (do^{4}-di^{4})}{32L}