Define cantilever beam.

Short Answer:

A cantilever beam is a type of beam that is fixed rigidly at one end and free at the other end. The fixed end resists both bending moment and shear force, while the free end carries the applied load. It bends under load, developing internal stresses to balance the applied forces and moments.

Cantilever beams are commonly used in balconies, bridges, overhanging roofs, and aircraft wings where one end must remain free. They are simple to construct and provide strong resistance against bending and deflection due to their fixed support.

Detailed Explanation:

Cantilever Beam

A cantilever beam is a structural member that is fixed at one end and free at the other end. It is one of the most widely used beam types in engineering structures. The fixed end of the beam is rigidly attached to a wall, column, or support, which prevents movement and rotation. The free end, however, can deflect or move when subjected to external loads.

The fixed support provides both vertical and horizontal reaction forces and a moment reaction to resist the applied loads. The cantilever beam is mainly subjected to bending moments and shear forces along its length, which cause the beam to bend downward when loaded. This property makes cantilever beams suitable for overhanging and projecting structures.

Cantilever beams are widely used in bridges, balconies, signboards, canopies, and cranes because they provide a clear, unobstructed area below the structure, requiring support only at one end.

Construction and Support Condition

A cantilever beam consists of two parts — a fixed end and a free end.

  1. Fixed End:
    • The beam is rigidly embedded into a wall or column at one end.
    • The fixed end can resist vertical and horizontal forces and moments.
    • It prevents both rotation and translation of the beam at that end.
  2. Free End:
    • The other end of the beam is free and carries the load.
    • This end undergoes maximum deflection when loaded.
    • The bending moment is highest at the fixed support and zero at the free end.

Due to this arrangement, cantilever beams experience both bending moments and shear forces throughout their length. The bending moment is negative (hogging) near the fixed end, which causes the top fibers of the beam to be in tension and the bottom fibers in compression.

Loading Conditions on a Cantilever Beam

Cantilever beams can be subjected to various types of loads depending on their application. The most common loading conditions are:

  1. Concentrated or Point Load:
    • A single load acting at the free end or anywhere along the beam.
    • Creates a linear variation of shear force and bending moment.
    • Maximum bending moment occurs at the fixed end.
  2. Uniformly Distributed Load (UDL):
    • Load distributed evenly along the length of the beam.
    • Creates a parabolic bending moment diagram.
    • Commonly occurs in roofs or bridges where weight acts over the entire span.
  3. Uniformly Varying Load (UVL):
    • Load intensity varies from zero at the free end to a maximum at the fixed end.
    • Creates a cubic variation of bending moment.
  4. Combination of Loads:
    • In real-life structures, the beam may experience a combination of point loads and distributed loads.

Each type of loading affects the magnitude and shape of the shear force diagram (SFD) and bending moment diagram (BMD), which help engineers design safe and efficient beams.

Shear Force and Bending Moment in a Cantilever Beam

The shear force (V) and bending moment (M) in a cantilever beam vary along its length depending on the applied load type.

  1. For a Point Load (P) at the Free End:
    • Shear force,  (constant throughout the beam).
    • Bending moment,  (varies linearly from zero at the free end to maximum at the fixed end).

Maximum bending moment at the fixed end:

  1. For a Uniformly Distributed Load (w) per unit length:
    • Shear force,  (at the fixed end).
    • Bending moment,  (maximum at the fixed end).

Here, L is the length of the beam and w is the load intensity.

The maximum bending stress and deflection occur near the fixed end, so that portion of the beam must be designed for maximum strength and stiffness.

Deflection of Cantilever Beam

The deflection of a cantilever beam depends on its length, type of load, and material properties. The maximum deflection occurs at the free end.

  1. For Point Load (P) at Free End:
  1. For Uniformly Distributed Load (w):

where,

  • E = Modulus of Elasticity,
  • I = Moment of Inertia of beam section,
  • L = Length of beam.

These formulas show that deflection increases rapidly with length and load, so cantilever beams must be properly designed to avoid excessive bending.

Advantages of Cantilever Beam

  1. No Intermediate Supports:
    • Requires support only at one end, leaving the other end free and clear.
  2. Simple and Economical Design:
    • Easy to construct and suitable for overhanging parts like balconies.
  3. Aesthetic and Functional:
    • Provides open space below the beam without obstruction.
  4. High Strength at Fixed End:
    • The fixed support provides strong resistance to bending and shear.
  5. Versatile Applications:
    • Suitable for bridges, roofs, aircraft wings, and signboards.

Disadvantages of Cantilever Beam

  1. High Bending Moment at Fixed End:
    • Requires stronger support and reinforcement.
  2. Deflection at Free End:
    • The free end experiences maximum deflection under load.
  3. Complex Design for Long Spans:
    • For longer lengths, deflection control and stability become challenging.
  4. Prone to Vibration:
    • Susceptible to vibration due to external forces like wind or moving loads.

Despite these limitations, cantilever beams are highly preferred in modern construction because of their versatility and architectural benefits.

Applications of Cantilever Beam

  1. Balconies and Canopies: Extend from building walls without external supports.
  2. Bridges: Used in cantilever bridge designs for long spans.
  3. Aircraft Wings: Act as cantilever beams projecting from the fuselage.
  4. Signboards and Traffic Signals: Supported at one end, free on the other.
  5. Roof Projections: Used in overhanging roof structures and shading devices.
Conclusion

A cantilever beam is a beam fixed at one end and free at the other, designed to carry loads acting perpendicular to its axis. It develops both bending moment and shear force, with maximum stress occurring at the fixed end. Cantilever beams are used in structures requiring free projections like balconies, bridges, and canopies. They provide strength, stability, and functional design with only one support, making them an essential element in mechanical and structural engineering.