Torsion of Circular Shafts in Strength of Materials

Torsion in circular shafts involves the application of a twisting moment or torque that induces shear stresses distributed over the shaft's cross section. These stresses vary linearly with the radius from the center, reaching a maximum at the outer surface. The shear stress $tau$ at any radius $r$ is given by $tau = \frac{T r}{J}$, where $T$ is the applied torque and $J$ is the polar moment of inertia of the cross section. For solid shafts, $\displaystyle J = \frac{\pi R^{4}}{2}$, and for hollow shafts, , where is the outer radius and the inner radius. The angle of twist over a shaft length is calculated by , where represents the shear modulus of the shaft material. Understanding these relationships is vital for designing shafts to transmit torque reliably without failure, balancing material usage and strength. Correctly predicting shear stress distribution and angle of twist helps prevent mechanical failure, optimize shaft dimensions, and select appropriate materials based on shear modulus.