Bending moment diagram for a cantilever beam carrying a uniformly distributed load over its length

Bending moment diagram shows the variation of bending moment along the length of the beam. The shape of the bending moment diagram for a cantilever beam carrying a uniformly distributed load over its length is a parabola.

Explanation:

When a cantilever beam carries a uniformly distributed load over its length, the load distribution is uniform throughout the length of the beam. Due to this uniform load distribution, the bending moment at any point along the beam length will also be uniform. The bending moment at any point is given by the product of the load and the distance of that point from the free end of the beam.

The bending moment at the free end of the beam is zero and it increases linearly towards the supported end. The maximum bending moment occurs at the supported end of the beam, where the bending moment is equal to the product of the total load and the length of the beam.

The shape of the bending moment diagram for a cantilever beam carrying a uniformly distributed load over its length is a parabola. The equation of the bending moment diagram is given by:

M = (wLx - wx^2/2)

where M is the bending moment, w is the load per unit length, L is the length of the beam, and x is the distance from the free end of the beam.

As x increases from zero to L, the bending moment increases linearly from zero to a maximum value at x = L. After that, the bending moment decreases linearly to zero at the supported end of the beam.

Conclusion:

Thus, the bending moment diagram for a cantilever beam carrying a uniformly distributed load over its length is a parabola. The knowledge of the bending moment diagram is essential in the design of beams, as it helps to determine the maximum bending moment and the required size of the beam.