When calculating the shear Force and the bending moment diagrams for more complex loading across discontinuities such as concentrated loads and moments. Simple methods are not enough. For the more complicated cases the use of singularity functions provide a convenient method.
A singularity function is expressed as
Where
n = any integer (positive or negative) including zero
a = distance on x axis along the beam,from the selected origin, identifying the location of the discontinuity.
| If n >0 and the expression inside the angular brackets is positive then fn(x) = (x-a)n the expression is a normal algbraic formula |
| If n > 0 and the expression inside the angular brackets is negative then fn(x) = 0 |
| If n < 0 then fn = 1 for x = a and fn(x) = 0 otherwise |
| If n = 0 then fn = 1 for x >= a and fn(x) = 0 otherwise |
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| Unit Singularity Function |
Singularity Function as used |
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Example of using singularity functions for a simply supported beam
The requirement is to obtain the Shear load, moment, slope and deflection anywhere along the beam as shown
The equations above can be used to determine the shear load, moment, slope and deflection for the beam from x = 0 to x = L