Bending of a rod by couples applied at its ends:

The rod is placed horizontally and is bent by applying a couple at its end as shown in the figure. In the small bending limit, the rod is stretched in the convex side, whereas in the concave side it is under compression. The neutral surface lying in the plane separates these two regions.
Here by small bending we mean that the deformation as well as the strain is small. Similar to bending of plates, external forces on the sides of the rod are small with respect to the internal stresses, so that at its sides, we have which ultimately results in that the all the components of the stress tensor except is zero. Let there be a small element lying along the axis at a distance from the neutral surface. On bending the length of this element becomes . The elements and lie on arcs of radii and respectively. Then it can be easily shown that

(12.1)

so that the relative expansion is . We now find as . The resultant traction over any cross-section is and it is equal to zero if the axis coincides with the line of centroid of the normal section. The component of the couple about the axis vanishes. However, that about the axis is nonzero and is or where is the moment of inertia of the section about an axis through its centroid parallel to the axis. Two more components of the strain tensor besides are non-zero: they are,

(12.2)