Maximum Bending Moment in a Beam

When a beam is subjected to a couple M at its center, it results in a maximum bending moment. The bending moment is the internal moment that causes bending in the beam. To understand how the maximum bending moment is determined, let's break down the concept step by step.

1. Definition of Bending Moment
The bending moment is the algebraic sum of the moments about a particular point in a beam. It is calculated by multiplying the force applied at a particular distance from the point of interest.

2. Bending Moment in a Simply Supported Beam
In a simply supported beam, the maximum bending moment occurs at the center of the beam. This is due to the symmetric nature of the loading condition. The bending moment gradually decreases from the center towards the ends of the beam.

3. Effect of Couple on Bending Moment
When a couple M is applied at the center of a beam, it creates a moment that tends to twist the beam. This moment is called the couple moment or torsional moment. The couple moment increases the bending moment at the center of the beam.

4. Calculation of Maximum Bending Moment
To calculate the maximum bending moment, we need to consider the combined effect of the couple moment and the applied loads on the beam.

- First, we determine the bending moment due to the applied loads using the principles of statics and structural analysis.
- Then, we calculate the bending moment due to the couple moment at the center of the beam.
- Finally, we add the bending moments from the applied loads and the couple moment to determine the maximum bending moment at the center of the beam.

5. Visual Representation
To visually represent the concept, a graph or diagram of the bending moment along the length of the beam can be plotted. The bending moment will be zero at the ends of the beam and will reach its maximum value at the center, where the couple moment is applied.

In conclusion, when a beam is pinned at both ends and subjected to a couple moment at its center, the maximum bending moment occurs at the center of the beam. This is due to the symmetric loading condition and the combined effect of the applied loads and the couple moment. Understanding the concept of bending moment is crucial in the design and analysis of beams in civil engineering.