Statically Indeterminate Torsion

A torsional system is statically indeterminate when equilibrium alone cannot determine all reaction torques. Additional equations follow from compatibility of rotations.

Three equation groups

1. Equilibrium: \(\sum M_x=0\).
2. Constitutive relation: \(d\varphi/dx=T/(GJ)\).
3. Compatibility: specified sections have equal rotation or zero relative rotation.

Shaft fixed at both ends

For an applied torque between two fixed supports, reaction torques satisfy equilibrium but require one additional condition:

\[\varphi_{AB}=\sum_i\frac{T_iL_i}{G_iJ_i}=0.\]

After finding the reactions, construct the final torque diagram and check strength and stiffness.

Composite coaxial shafts

Rigidly connected coaxial shafts undergo the same angle of twist, while the applied torque is shared between components. The share depends on their torsional stiffnesses \(GJ/L\).

Procedure

1. Introduce unknown reaction torques.
2. Write equilibrium.
3. Express internal torque on every segment.
4. Write rotation compatibility.
5. Solve reactions, construct the torque diagram, and complete design checks.