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Introduction
The pure torsion refers to a torsion of a prismatic bar subjected to torques acting only at the ends. While the non uniform torsion differs from pure torsion in a sense that the bar / shaft need not to be prismatic and the applied torques may vary along the length.
Here the shaft is made up of two different segments of different diameters and having torques applied at several cross sections. Each region of the bar between the applied loads between changes in cross section is in pure torsion, hence the formula's derived earlier may be applied. Then form the internal torque, maximum shear stress and angle of rotation for each region can be calculated from the relation
The total angle to twist of one end of the bar with respect to the other is obtained by summation using the formula
If either the torque or the cross section changes continuously along the axis of the bar, then the (summation can be replaced by an integral sign ( ∫ ). i.e We will have to consider a differential element.
After considering the differential element, we can write 
Substituting the expressions for Tx and Jx at a distance x from the end of the bar, and then integrating between the limits 0 to L, find the value of angle of twist may be determined.
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