When considering a beam under continuous loading, the shear force at any point is represented by mathematical functions.
However, when the beam experiences discontinuous loading, different functions are required to accurately represent the shear force in various parts of the beam.
In such cases, singularity functions allow the representation of shear force with a single mathematical expression despite the varying loading conditions.
To derive the singularity functions, a free-body diagram of the beam is drawn and is conceptually cut at specific points. Then, the singularity function representing the shear force in each beam portion is determined.
Applying the convention that angle brackets or Macaulay's brackets are replaced with parentheses when x is greater than or equal to l, and with zero when x is less than l, these singularity functions can be differentiated or integrated like ordinary mathematical expressions.
The singularity functions are plotted for visual representation. Most beam loadings can be broken down into basic loadings, and the functions for shear force can be obtained by adding the corresponding functions for each loading.
