Simple Beam - Uniformly Distributed Load

A simple beam is supported at both ends, with a uniformly distributed load (UDL) applied over its entire length. This type of loading is common in structural and mechanical applications, where loads such as self-weight, snow, or evenly distributed forces act along the beam's span.

Key Concepts

• Simple Beam: A beam with supports at both ends, allowing rotation but preventing vertical displacement.
• Uniformly Distributed Load (UDL): A constant load applied per unit length along the entire span of the beam.
• Shear Force: The shear force varies linearly along the beam, with maximum values at the supports.
• Bending Moment: The bending moment is parabolic, reaching its maximum at the center of the beam.
• Deflection: The deflection is greatest at the center, forming a symmetric curve along the beam.

Behavior of the Simple Beam

1. Reaction Forces:
   • The total equivalent force of the UDL is its intensity multiplied by the beam length.
   • The reactions at the supports are equal, each carrying half of the total load.
2. Shear Force Diagram:
   • The shear force decreases linearly from one support to the other.
   • The maximum shear force occurs at the supports and is equal to half the total applied load.
3. Bending Moment Diagram:
   • The bending moment follows a parabolic shape, with a maximum at the center.
   • The maximum bending moment is given by \( M_{\text{max}} = \frac{wL^2}{8} \), where \( w \) is the load intensity and \( L \) is the beam length.
4. Deflection: The maximum deflection occurs at the center and is given by: \[ \delta_{\text{max}} = \frac{5wL^4}{384EI} \] where \( E \) is the modulus of elasticity and \( I \) is the moment of inertia of the beam cross-section.

Applications

• Structural Engineering: Used in floor beams, bridge spans, and roofing structures subject to uniform loads.
• Mechanical Systems: Found in conveyor belts, machine frames, and automotive chassis designs.
• Construction: Relevant for analyzing beams in buildings that carry uniform loads such as flooring, ceilings, or distributed equipment weight.

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