Cantilever Beam - Uniform Load Partially Distributed

A cantilever beam is a structural element that is fixed at one end and free at the other. When a uniformly distributed load is applied to only a part of the beam, it causes varying shear force, bending moment, and deflection along the length of the beam. In this case, the load is applied over a section of the beam, starting from the fixed end and extending for a length \( a \), rather than the full span of the beam.

Key Concepts

• Uniform Load: A constant load intensity, \( w \), applied uniformly along a section of the beam, extending a length \( a \) from the fixed end. The load intensity remains constant within this section of the beam.
• Fixed End: The end of the beam that is rigidly attached to a support, preventing both rotation and translation.
• Free End: The unsupported end of the beam, where the beam is free to move and no load is applied.
• Shear Force: The shear force is influenced by the uniform load, and it increases from the free end to the fixed end, with a sudden change in magnitude at the end of the loaded section.
• Bending Moment: The bending moment increases linearly under the applied uniform load and reaches its maximum at the fixed end of the beam.
• Deflection: The deflection increases due to the uniform load, with the maximum deflection typically occurring closer to the free end of the beam.

Behavior of the Cantilever Beam

1. Reaction Forces:
   • At the fixed end, the total equivalent load from the uniform load is equal to \( w \cdot a \), where \( w \) is the load intensity and \( a \) is the length of the loaded section.
   • A reaction moment \( M_A \) is generated at the fixed end to counteract the effect of the applied load.
2. Shear Force Diagram:
   • The shear force is constant within the loaded region and then changes after the load ends, depending on the position along the beam.
   • The maximum shear force occurs at the fixed end and is given by \( V_A = w \cdot a \).
3. Bending Moment Diagram:
   • The bending moment increases linearly in the region of the uniformly distributed load, and reaches its maximum value at the fixed end.
   • After the uniform load section ends, the bending moment remains constant along the rest of the beam.
   • The maximum bending moment at the fixed end is given by \( M_A = \frac{w \cdot a^2}{2} \).
4. Deflection: The deflection is influenced by both the uniform load intensity and the length of the loaded portion. The deflection at any point along the beam can be calculated using standard beam deflection formulas, and the maximum deflection occurs near the free end. The deflection at the free end is given by: \[ \delta_{\text{max}} = \frac{w \cdot a^4}{8EI} \] where \( E \) is the modulus of elasticity and \( I \) is the moment of inertia of the beam's cross-section.

Applications

• Structural Engineering: Common in beams subjected to localized loads, such as beams supporting equipment or structures near one end.
• Mechanical Systems: Found in components where loads are applied over a portion of the beam's length, such as cantilever arms supporting machinery or devices.
• Construction: Used in analyzing beams with loads applied at specific sections, such as beams carrying partition walls or localized equipment at one end.

Formula

Definitions