Internal Torque and Torque Diagrams

Internal torque \(T\) is the stress resultant developed at a shaft cross-section under torsional loading. It is found by the method of sections and rotational equilibrium of either cut segment.

Applied and internal torques

Applied torque may be produced by a motor, coupling, pulley, gear, or force couple. After an imaginary cut, the internal torque balances the algebraic sum of external torques acting on one side:

\[T(x)=-\sum M_x.\]

Sign convention

Choose a positive direction once and use it consistently. The right-hand rule about the longitudinal axis is convenient. Internal torques on opposite cut faces have opposite directions.

Torque diagram

On a segment without distributed torsional loading, \(T\) is constant. A concentrated applied torque produces a jump of equal magnitude. For distributed torque \(m_t(x)\):

\[\frac{dT}{dx}=-m_t(x).\]

Procedure

1. Show all applied and reaction torques.
2. Find unknown reactions from \(\sum M_x=0\).
3. Divide the shaft into loading intervals.
4. Cut each interval and calculate \(T(x)\).
5. Draw the diagram and check jumps and overall equilibrium.