Cantilever Beam - Couple Moment \( M_o \) at Any Point

A cantilever beam is a structural element fixed at one end and free at the other. When a couple moment \( M_o \) is applied at any point along the beam, it induces specific effects such as deflection, bending moment, and shear force. Unlike a concentrated load, a couple moment does not produce a resultant force but instead creates pure bending.

Key Concepts

• Couple Moment \( M_o \): A moment (torque) applied at a specific point, consisting of two equal and opposite forces. Measured in units of force × distance (e.g., N·m or lb·ft).
• Fixed End: The end rigidly attached to a support, resisting rotation and translation.
• Free End: The unsupported end that experiences deflection due to the applied moment.
• Shear Force: Zero throughout the beam because the couple moment does not create any shear force.
• Bending Moment: Varies along the beam and changes abruptly at the point of the couple moment.
• Deflection: The displacement of the beam, maximum at the free end.

Behavior of the Cantilever Beam

1. Reaction Forces:
   • At the fixed end, the beam generates a reaction moment \( M_R = M_o \).
   • No vertical reaction force is produced.
2. Shear Force Diagram: The shear force is zero throughout the beam.
3. Bending Moment Diagram:
   • Constant between the fixed end and the point of the couple moment.
   • Changes abruptly by \( M_o \) at the point of application.
   • Remains constant beyond the point of application.
4. Deflection: Maximum at the free end, calculated using beam deflection formulas.

Applications

• Engineering Design: Used in structures subjected to torsional or rotational loads.
• Mechanical Systems: Applied in shafts, levers, and supports.
• Construction: Used to calculate the stability of cantilevered structures under rotational forces.

Formula

Definitions