Cantilever Beam - Concentrated Load \( P \) at Free End

A cantilever beam is a structural element fixed at one end and free at the other. When a concentrated load \( P \) is applied at the free end, it induces shear force, bending moment, and deflection along the beam. This is a common loading condition that results in the maximum possible deflection and bending moment at the fixed support.

Key Concepts

• Concentrated Load \( P \): A single force applied at the free end of the beam, measured in units of force (e.g., N or lb).
• Fixed End: The end rigidly attached to a support, resisting both rotation and translation.
• Free End: The unsupported end where the load is applied, experiencing the maximum deflection.
• Shear Force: Constant along the entire length of the beam.
• Bending Moment: Varies linearly along the beam, reaching its maximum at the fixed end.
• Deflection: Maximum at the free end, causing the beam to bend downward.

Behavior of the Cantilever Beam

1. Reaction Forces:
   • At the fixed end, the beam generates a reaction force \( R_A = P \) to balance the external load.
   • A reaction moment \( M_A = P \cdot L \) is developed, where \( L \) is the length of the beam.
2. Shear Force Diagram:
   • The shear force remains constant and equal to \( P \) throughout the entire beam.
3. Bending Moment Diagram:
   • The bending moment varies linearly from zero at the free end to a maximum of \( M_A = P \cdot L \) at the fixed end.
4. Deflection: Maximum at the free end, calculated using the formula: \[ \delta_{\text{max}} = \frac{P L^3}{3EI} \] where \( E \) is the modulus of elasticity and \( I \) is the moment of inertia of the beam cross-section.

Applications

• Structural Engineering: Common in overhanging balconies, diving boards, and cantilevered beams in buildings.
• Mechanical Systems: Used in robotic arms, levers, and crane booms.
• Construction: Helps determine the deflection and bending moment for designing safe cantilever structures.

Formula

Definitions