# Torsion

## Torsion

Consider a bar to be rigidly attached at one end and twisted at the other end by a torque or twisting moment *T* equivalent to *F* × *d*, which is applied perpendicular to the axis of the bar, as shown in the figure. Such a bar is said to be in torsion.

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## Torsional Shearing Stress, τ

For a solid or hollow circular shaft subject to a twisting moment *T*, the torsional shearing stress τ at a distance ρ from the center of the shaft is

where *J* is the polar moment of inertia of the section and *r* is the outer radius.

**For solid cylindrical shaft:**

$J = \dfrac{\pi}{32} D^4$

$\tau_{max} = \dfrac{16T}{\pi D^3}$

**For hollow cylindrical shaft:**

$J = \dfrac{\pi}{32}(D^4 - d^4)$

$\tau_{max} = \dfrac{16TD}{\pi(D^4 - d^4)}$

## Angle of Twist

The angle θ through which the bar length *L* will twist is

where *T* is the torque in N·mm, *L* is the length of shaft in mm, *G* is shear modulus in MPa, *J* is the polar moment of inertia in mm^{4}, *D* and *d* are diameter in mm, and *r* is the radius in mm.

## Power Transmitted by the Shaft

A shaft rotating with a constant angular velocity ω (in radians per second) is being acted by a twisting moment *T*. The power transmitted by the shaft is

where *T* is the torque in N·m, *f* is the number of revolutions per second, and *P* is the power in watts.