The Maximum Bending Moment in a Simply Supported Beam

In a simply supported beam, the maximum bending moment occurs at the center of the beam when a central concentrated load is applied. However, when the same load is distributed uniformly over the entire span, the maximum bending moment is reduced. Let's understand why.

The Effect of Load Distribution on Bending Moment

When a central concentrated load is applied to a simply supported beam, the load is concentrated at a single point, resulting in a high bending moment at the center. This is because the load is acting at the greatest distance from the supports, causing the maximum deflection of the beam.

However, when the same load is distributed uniformly over the entire span, the load is spread out, resulting in a reduction of the bending moment. This is because the load is now acting at a shorter distance from the supports, causing a smaller deflection of the beam.

Calculating the Maximum Bending Moment

To understand why the maximum bending moment is M/2 when the load is distributed uniformly, we can consider the equilibrium condition of the beam. In a simply supported beam, the bending moment at any point is given by the equation:

M = (wL^2)/8

Where:
M is the bending moment
w is the load per unit length (uniform load)
L is the length of the beam

When the load is distributed uniformly, the load per unit length (w) is equal to the total load (W) divided by the length of the beam (L). Therefore, the equation for the bending moment becomes:

M = (WL^2)/8L

Simplifying the equation:

M = (WL)/8

Since the maximum bending moment occurs at the center of the beam, we can consider half of the beam length (L/2) in the equation. Substituting L/2 for L, we get:

M = (W(L/2))/8

Simplifying further:

M = (W/2)(L/8)

M = (W/2)(L/4)

M = (W/2)(L/4)

M = (W/2)(L/4)

M/2 = (W/2)(L/4)

Therefore, the maximum bending moment when the load is distributed uniformly is M/2.

Conclusion

In a simply supported beam, a central concentrated load causes a maximum bending moment at the center. However, when the same load is distributed uniformly over the entire span, the maximum bending moment is reduced to M/2. This reduction occurs because the load is acting at a shorter distance from the supports, resulting in a smaller deflection of the beam.