Consider a simply supported beam carrying a uniformly distributed load from its midpoint to its right-hand support.

To determine the bending moment in beams using singularity functions, the free-body diagram of the beam is drawn by replacing the distributed load with an equivalent concentrated one.

The moments are then summed about the right-hand support to obtain the total moment equation to determine the magnitude of the reaction force.

Next, the beam is cut at a point between the left-hand support and the mid-point. From its free-body diagram, the bending moment is expressed by a specific function by considering the interval.

Now, the beam is cut at a point between the mid-point and the right-hand support. The free-body diagram of this portion is drawn, replacing the distributed load with an equivalent concentrated load, and the bending moment is expressed by a different function by considering the interval.

These two functions are combined into a single expression representing the bending moment at any beam point. The expression within the angle brackets for different conditions is called the singularity function.