Torsion occurs when a twisting moment or torque is applied to a structural or mechanical member. This module examines the effects of torsion, particularly on circular solid and hollow shafts, which are ubiquitous in mechanical and structural engineering as axles, drive shafts, and structural columns supporting eccentric loads.

When a circular shaft is subjected to a torque ($T$), shear stresses develop within the material to resist the twisting. Crucially, in a circular shaft, this shear stress varies linearly from exactly zero at the center of the shaft to an absolute maximum at its extreme outer surface.

The maximum torsional shear stress ($\tau_{max}$) at the outer fibers is given by the general torsion formula:

Where:

For a solid circular shaft, the polar moment of inertia ($J$) is a geometric property representing the cross-section's resistance to twisting:

For a hollow circular shaft with outer diameter $D$ and inner diameter $d$, the material in the center (where stress would be lowest anyway) is removed, making it highly efficient. The polar moment of inertia is:

Just as an axial load causes longitudinal deformation (elongation), a torsional load causes angular deformation, known as the angle of twist ($\theta$). This is the angle by which one end of the shaft rotates relative to the other end.

The formula for the angle of twist over a length $L$ is:

Where:

Beyond standard solid and hollow circular shafts, engineers frequently encounter complex torsional scenarios in practical design. Two highly common applications are flanged bolt couplings (used to connect two separate shafts) and thin-walled tubes.

When two shafts are connected end-to-end via a bolted flange, the entire torsional moment is transmitted from one shaft to the other entirely through the shear resistance of the bolts.

If $n$ is the total number of bolts, $A_b$ is the cross-sectional area of one bolt, $d$ is the bolt diameter, $R$ is the radius of the bolt circle (the distance from the center of the shaft to the center of the bolts), and $\tau$ is the allowable shear stress in the bolts, the maximum torque that the coupling can safely transmit is governed by:

This is simply stating that Torque = (Total Shear Force Resisted by all Bolts) $\times$ (Lever Arm from Center).

When a twisting moment is applied to a non-circular thin-walled tube (like a rectangular aluminum extrusion or a closed steel box section), the shear stress is no longer distributed linearly from the center as it is in a solid circular shaft. Instead, it is distributed as a uniform shear flow ($q$) traveling around the perimeter of the cross-section.

The relationship between the applied torque $T$ and the shear flow $q$ is defined by Bredt's formula:

Where:

Since shear flow $q = \tau \cdot t$ (where $t$ is the wall thickness at any point), the average shear stress at any specific point along the perimeter is:

Circular shafts are widely used in mechanical engineering to transmit power from a motor to a machine. The fundamental physical relationship between mechanical power ($P$), applied torque ($T$), and rotational speed ($\omega$ or $N$) is:

Where angular velocity $\omega$ (in radians per second) is related to rotational speed $N$ (in revolutions per minute, RPM) by $\omega = \frac{2\pi N}{60}$.

Substituting $\omega$ into the power equation yields the common engineering formula:

Where:

A closely coiled helical spring subjected to an axial load ($P$) behaves fundamentally as a torsional member. The wire forming the spring experiences a twisting moment as the coils try to flatten out under the load.

The maximum shear stress ($\tau_{max}$) in a helical spring is a combination of direct shear stress and torsional shear stress. To accurately calculate this, we use Wahl's formula, which introduces a correction factor ($K_W$) to account for the curvature of the inner face of the coil.

Where:

The deflection ($\delta$) of a closely coiled helical spring under an axial load $P$ with $n$ active coils is given by:

Where: