19.1 : 轴中的应力

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.