Pure torsional loading (couple only, no bending) produces a shearing stress in a shaft with the magnitude of the stress, in any cross section, being proportional to the distance from the center of the shaft (Fig. 6-7). This follows from the fact that the deformation at any point equals-- ? d ?, and since stress is proportional to strain the stress increases from zero at the center to a maximum at the outside. Let S _s be the maximum shearing stress at the outer fiber. Then S _s/r will be the stress one meter from the center and ( ?/r) S _s will be the stress at a distance ? from the center. Force equals stress times area, so the force resisting the applied torque, due to the area dA equals ( ?/r)S _s dA; the resisting moment about the axis of the bar due to this force is equal to ( ? ²/r)S _s dA, and (S _s/r) ? ? ² dA is the total moment about the axis due to all the internal shearing forces. The quantity ? ? ² dA is, we recall from Mechanics (see Handbook), the moment of inertia of the cross-sectional area about the axis through its center or the polar moment of inertia of the area, which quantity is ordinarily represented by the symbol J. This gives us the relationship that torque = (S