Simple Beam - Sinusoidal Distributed Load

A simple beam is supported at both ends, with a sinusoidal distributed load applied along its length. The load follows a sine wave pattern, with the intensity of the load varying along the beam. The load typically starts at zero at one support, increases to a maximum at the center of the beam, and then decreases back to zero at the other support. This load distribution results in complex shear forces, bending moments, and deflections along the beam.

Key Concepts

• Simple Beam: A beam supported at both ends, with no intermediate supports.
• Sinusoidal Load: A load that follows a sine wave pattern, where the intensity of the load varies sinusoidally along the length of the beam, starting from zero at the ends and peaking at the center.
• Shear Force: The shear force varies along the beam as a result of the sinusoidal load distribution. It starts at zero at the supports and changes as the intensity of the load changes along the beam.
• Bending Moment: The bending moment follows a sinusoidal pattern as well, increasing from zero at the supports to a maximum at the center of the beam, then decreasing back to zero at the other support.
• Deflection: The deflection of the beam is maximum at the center, where the load intensity is highest. The deflection follows the curvature of the applied load distribution.

Behavior of the Simple Beam

1. Reaction Forces:
   • The total load on the beam can be calculated by integrating the sinusoidal load distribution over the length of the beam. The reactions at the supports are determined by summing the vertical forces and applying equilibrium conditions.
2. Shear Force Diagram:
   • The shear force diagram reflects the sinusoidal nature of the load distribution. It shows the variations in shear force along the length of the beam, starting from zero at the supports and varying along the beam.
3. Bending Moment Diagram:
   • The bending moment diagram follows a similar sinusoidal pattern, with a maximum value at the center of the beam, where the load is greatest. The moment decreases symmetrically towards the supports.
4. Deflection: The deflection is greatest at the center of the beam, where the bending moment is largest. The deflection follows the general shape of the load distribution and can be determined using deflection formulas for sinusoidal loading.

Applications

• Structural Engineering: Sinusoidal loading is often used to model periodic forces, such as vibrations or waves that affect beams and other structures. It is common in modeling dynamic loads like wind or earthquake forces.
• Mechanical Systems: Found in components subjected to periodic loading, such as beams in machines or parts under oscillating forces.
• Construction: Used in the analysis of beams and structures subjected to dynamic loads, such as those found in bridges or buildings in areas with periodic loading conditions.

Formula

Definitions