Singularity Functions for Shear

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Mechanical Engineering
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JoVE Core Mechanical Engineering
Singularity Functions for Shear

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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.

Singularity Functions for Shear

In structural analysis, singularity functions are crucial in simplifying the representation of shear forces in beams under discontinuous loading. These functions describe discontinuous  variations in shear force across a beam with varying loads by using a single mathematical expression, regardless of the complexity of the loading conditions. The singularity functions are derived from creating a free-body diagram of the beam and then making conceptual cuts at specific points to examine the shear force in each section. This is defined by

Where W and L are the width and length of the beam, respectively.  Macaulay's brackets < > are an essential feature of singularity functions. These brackets help evaluate the function based on the position along the beam about specific points of interest. The notation adjusts the function's value to account for the beam's condition at different sections, enabling these functions to be treated like standard mathematical expressions for purposes of differentiation and integration.

For example, when dealing with a point load placed at a certain point along the beam, singularity functions allow for a straightforward representation of the abrupt change in shear force at that location. By breaking down complex beam loadings into simpler components, the overall shear force across the beam can be easily determined by combining the functions associated with each type of loading.