21.4:
Singularity Functions for Bending Moment
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.
21.4:
Singularity Functions for Bending Moment
Singularity functions simplify the representation of bending moments in beams subjected to discontinuous loading, allowing the use of a single mathematical expression. For a supported beam AB, with uniform loading from its midpoint M to the right side end B, the approach involves conceptual 'cuts' at specific points to determine the bending moment in each segment. By cutting the beam at a point between A and M, the bending moment for the segment before reaching midpoint M is represented using a particular function.
Another cut at a point between M and B allows for the bending moment for the segment from midpoint M to the end of the beam to be described by a different function. The key to simplifying the representation is combining these functions into a single expression that adapts based on the position along the beam.
Where w0 is the distributed load applied over the length from M to the end of the beam. The expression is formed by including the second function in calculations only for positions beyond midpoint M, effectively using a conditional approach to manage the discontinuity. Furthermore, the distribution of the load along the beam, and the resulting shear force, can also be depicted using singularity functions. This method, often employing Macaulay's brackets for representation, streamlines the calculation of bending moments in beams with varying loading conditions.