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Section 2-08 Singularity Functions

The five singularity functions defined in Table 2-1 constitute a useful and easy means of integrating across discontinuities. By their use, general expressions for shear force and bending moment in beams can be written when the beam is loaded by concentrated forces or moments. As shown in the table, the concentrated moment and force functions are zero for all values of x except the particular value of x = a. The unit step, ramp, and parabolic functions are zero only for values of x that are less than a. The integration properties shown in the table constitute a part of the mathematical definition too.

The functions are presented in text form for copy/paste use as follows

     Concentrated
     Moment           <x-a>^-2 = 0 for x<>a
             INTEGRAL(-infinity,x, <x-a>^-2 * dx) = <x-a>^-1

     Concentrated 
     Force            <x-a>^-1 = 0 for x<>a
             INTEGRAL(-infinity,x, <x-a>^-1 * dx) = <x-a>^0
       
     Unit Step        <x-a>^0 = 0 for x<a
                      <x-a>^0 = 1 for x=>a
             INTEGRAL(-infinity,x, <x-a>^0 * dx) = <x-a>^1

     Ramp             <x-a>^1 = 0 for x<a
                      <x-a>^1 = x-a for x=>a
             INTEGRAL(-infinity,x, <x-a>^1 * dx) = 1/2*<x-a>^2

     Parabolic        <x-a>^2 = 0 for x<a
                      <x-a>^1 = (x-a)^2 for x=>a
             INTEGRAL(-infinity,x, <x-a>^2 * dx) = 1/3*<x-a>^3

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