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

• Geometry of deformation
From the angles of rotation at $\mathrm{A}$, $\mathrm{B}$, and $\mathrm{C}$, we will deduce the relationship between the twist angles in shafts $\left(1\right)$ and $\left(2\right)$.

• 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 ${\mathrm{T}}_2$.