Singularity functions are a class of discontinuous functions that contain singularities, i.e. they are discontinuous at their singular points. Singularity functions have been heavily studied in the field of mathematics under the alternative names of generalized functions and distribution theory. The functions are notated with brackets, as
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where: δ(x) is the Dirac delta function, also called the unit impulse. The first derivative of δ(x) is also called the unit doublet. The function
Integration
Integrating
Example beam calculation
The deflection of a simply supported beam as shown in the diagram, with constant cross-section and elastic modulus, can be found using Euler-Bernoulli beam theory. Here we are using the sign convention of downwards forces and sagging bending moments being positive.
Load distribution:
Shear force:
Bending moment:
Slope:
Deflection:
The boundary condition u=0 at x=4m allows us to solve for c=-7Nm2