19.1:
Stresses in a Shaft
A shaft PQ is twisted when subjected to equal and opposite torques on either side.
Consider a section perpendicular to the shaft's axis through an arbitrary point R. The free-body diagram of portion QR shows the shearing forces exerted by portion PR on QR as the shaft twists.
Applying the equilibrium equations to portion QR, it can be shown that the shearing forces within the section are related to the internal torque. Here, r represents the perpendicular distance from the shaft's axis to the shearing force.
Now consider a small area element of the shaft where the shearing force can be expressed as the product of shearing stress and the area element.
By substituting this relation, the expression for torque is obtained in terms of shearing stress.
This relation must be satisfied by the shearing stresses in any cross-section of the shaft. However, it does not provide information about the distribution of these stresses in the cross-section.
The distribution of shearing stresses in an elastic shaft is indeterminate by statics alone and requires deformation analysis.
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.