19.1 : Stresses in a Shaft

The shaft PQ is subjected to a twisting force when equal and opposite torques are applied on either side. A section that cuts perpendicular to the shaft's axis at any arbitrary point R is examined to understand this. When the free-body diagram of the QR segment is analyzed, it reveals the shearing forces exerted by the PR portion onto the QR segment as the shaft experiences twisting.

Applying equilibrium conditions to the QR segment establishes that the internal shearing forces within the section directly correlate with the internal torque. Here, 'r' signifies the perpendicular distance from the axis of the shaft to the shearing force. Next, a small area element of the shaft is taken into account. The shearing force can be expressed as the multiplication of the shearing stress and the area element. Upon substituting this relation, an expression for torque in terms of shearing stress is derived.

This derived relation must hold true for the shearing stresses in any shaft cross-section. However, it does not provide insights into the distribution of these stresses across the cross-section. Lastly, it is important to note that the distribution of shearing stresses in an elastic shaft cannot be determined solely by statics. It requires deformation analysis for accurate determination.