Torsion of rods

You can deform a thin rod in a number of ways, e.g. you can bend it, twist it and you can both bend and twist it. In all these cases, even though the relative displacement of neighboring points may be small, i.e. the strain may be small but the overall deformation may still be large. We first consider here the problem of torsion of a long thin rod.

In torsional deformation, while the rod remains straight, each transverse element in it rotates through some angle relative to the one below it. Consequently, after the rod is twisted through some angle, any generator line parallel to the axis no linger remains straight but turns helical. The complete solution of the displacement should satisfy the following stress equilibrium relations

(15.1)

and the boundary conditions are:

on the side surface

and equal to end loads at .