Cantilever Beam - Load Increasing Uniformly to Fixed End

A cantilever beam is a structural element fixed at one end and free at the other. When a linearly increasing distributed load is applied along the beam, with the load intensity increasing from zero at the free end to a maximum \( w_0 \) at the fixed end, it creates varying shear force, bending moment, and deflection along the beam.

Key Concepts

• Linearly Increasing Load: A distributed load that starts from zero at the free end and increases to \( w_0 \) at the fixed end, expressed as \( w(x) = w_0 \frac{x}{L} \), where \( L \) is the beam length.
• Fixed End: The end rigidly attached to a support, resisting both rotation and translation.
• Free End: The unsupported end where the load intensity is zero.
• Shear Force: Varies non-linearly along the beam due to the gradually increasing load.
• Bending Moment: The moment increases more rapidly near the fixed end, reaching a maximum at that point.
• Deflection: The maximum deflection occurs at the free end and follows a curved shape due to the varying load.

Behavior of the Cantilever Beam

1. Reaction Forces:
   • At the fixed end, the total equivalent load acts as a concentrated force of magnitude \( \frac{w_0 L}{2} \).
   • A reaction moment \( M_A \) is developed to counteract the distributed load.
2. Shear Force Diagram:
   • The shear force follows a quadratic variation along the beam.
   • The maximum shear force occurs at the fixed end and is given by \( V_A = \frac{w_0 L}{2} \).
3. Bending Moment Diagram:
   • The bending moment increases cubically along the beam.
   • The maximum moment at the fixed end is given by \( M_A = \frac{w_0 L^2}{6} \).
4. Deflection: The deflection at the free end is given by: \[ \delta_{\text{max}} = \frac{w_0 L^4}{30EI} \] where \( E \) is the modulus of elasticity and \( I \) is the moment of inertia of the beam cross-section.

Applications

• Structural Engineering: Common in beams subjected to wind pressure or soil pressure.
• Mechanical Systems: Found in components experiencing variable loading, such as machine parts and turbine blades.
• Construction: Used in analysis of cantilevered structures under self-weight and other non-uniform loads.

Formula

Definitions