Formula

Cantilever Beam - Couple Moment \( M_o \) at Free End

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 the free end, it induces a pure bending effect without generating shear force. This loading condition primarily influences the bending moment distribution and deflection of the beam.

Key Concepts

• Couple Moment \( M_o \): A moment (torque) applied at the free end, measured in units of force × distance (e.g., N·m or lb·ft).
• Fixed End: The end rigidly attached to a support, resisting both rotation and translation.
• Free End: The unsupported end where the moment is applied, causing rotational displacement.
• Shear Force: Zero throughout the beam since the applied couple moment does not generate a resultant force.
• Bending Moment: The moment remains constant throughout the beam, equal to \( M_o \).
• Deflection: The beam experiences angular rotation at the free end without transverse displacement.

Behavior of the Cantilever Beam

1. Reaction Forces:
   • At the fixed end, the beam generates a reaction moment \( M_R = M_o \) to balance the applied couple moment.
   • No vertical reaction force is produced.
2. Shear Force Diagram: The shear force remains zero throughout the entire length of the beam.
3. Bending Moment Diagram:
   • The bending moment is constant along the beam and equal to \( M_o \).
   • Unlike other loading conditions, there is no variation in the moment distribution.
4. Deflection: The beam undergoes rotation at the free end, calculated using: \[ \theta_{\text{max}} = \frac{M_o L}{EI} \] where \( E \) is the modulus of elasticity, \( I \) is the moment of inertia, and \( L \) is the beam length.

Applications

• Structural Engineering: Used in beams and supports subjected to pure moments at their free ends.
• Mechanical Systems: Common in robotic arms, torque shafts, and rotating cantilevered components.
• Construction: Helps analyze the effects of end moments in overhanging structures.

Definitions