• Equilibrium
Draw a free-body diagram $\left(\mathrm{FBD}\right)$ of end $\mathrm{B}$ to determine the relationship of internal torques ${\mathrm{T}}_1$ and ${\mathrm{T}}_2$ to the applied torque $\mathrm{T}$.

• Geometry of deformation
Since the two shafts are bonded together, the angle of twist in both shafts must be identical.

• Torque-twist relationships
The relationship between internal torque and angle of twist will be expressed for each shaft.

• Compatibility equation
The torque-twist relationships will be substituted into the geometry of deformation relationship to derive a compatibility equation expressed in terms of the internal torques ${\mathrm{T}}_1$ and ${\mathrm{T}}_2$.