Fixed-Fixed Beam - Concentrated Load at Any Point

A **fixed-fixed beam** is a beam that is supported at both ends, with both ends rigidly fixed in place, preventing rotation and translation. When a **concentrated load** is applied at any point along the beam, it causes shear force, bending moment, and deflection that vary depending on the position of the load along the beam.

Key Concepts

• Concentrated Load: A load applied at a specific point along the beam. It acts as a point force, and its effect is localized to the point of application.
• Fixed-Fixed Beam: A beam that is supported at both ends, with both ends fixed, meaning that there is no rotation or displacement allowed at these supports.
• Shear Force: The shear force is a function of the applied concentrated load and varies along the beam. It changes direction at the point where the load is applied.
• Bending Moment: The bending moment is greatest at the point of load application and at the supports, and its distribution along the beam depends on the position of the applied load.
• Deflection: The deflection occurs due to the applied concentrated load and is maximized at the point of application. The beam deflects in response to this localized force.

Behavior of the Fixed-Fixed Beam

1. Reaction Forces:
   • The fixed supports at both ends of the beam develop reaction forces to balance the applied concentrated load.
   • The total reaction forces at the supports depend on the position of the applied load, and they are calculated using equilibrium equations for static analysis.
2. Shear Force Diagram:
   • The shear force is constant between the fixed supports and changes at the point of load application.
   • The shear force jumps by the magnitude of the applied concentrated load at the point of application.
   • The maximum shear force occurs near the supports, with the direction of the force depending on the position of the load.
3. Bending Moment Diagram:
   • The bending moment increases linearly from one support toward the point of load application, then decreases toward the other support.
   • The maximum bending moment occurs at the point of load application or at one of the fixed supports, depending on the load's location.
   • The bending moment is calculated using equilibrium equations and is affected by the magnitude and location of the applied load.
4. Deflection: The deflection due to the concentrated load can be calculated using beam deflection formulas. The deflection is highest at the point of application of the load, and it decreases toward the fixed supports. \[ \delta = \frac{P \cdot L^3}{48 E I} \] where \( P \) is the concentrated load, \( L \) is the length of the beam, \( E \) is the modulus of elasticity, and \( I \) is the moment of inertia of the beam's cross-section.

Applications

• Structural Engineering: Found in beams subjected to point loads, such as beams in bridges, buildings, and frames.
• Mechanical Systems: Used in components where localized loads are applied, such as shafts, arms, or supports in mechanical machinery.
• Construction: Relevant in the analysis of beams that carry point loads, such as beams that support walls or equipment in buildings.

Formula

Definitions