Consider a circular shaft of length L radius R, subjected to a torque T at one end (see Fig.1). Under the action of torque one end of the shaft rotates with respect to the fixed end by an angle dφ. Hence the strain energy stored in the shaft is,
Consider an elemental length ds of the shaft. Let the one end rotates by a small amount dφ with respect to another end. Now the strain energy stored in the elemental length is,
where, G is the shear modulus of the shaft material and is J the polar moment of area. Substituting for dφ from (3) in equation (2), we obtain
Now, the total strain energy stored in the beam may be obtained by integrating the above equation.
