Singularity function

Singularity functions or singularity brackets are a notation used to describe certain discontinuous functions. The functions are written as $\langle x-a\rangle ^{n}$ where n is an integer. They are defined as:

n	$\langle x-a\rangle ^{n}$
$<0$	${\frac {d^{|n+1|}}{dx^{|n+1|}}}\delta (x-a)\,$
-2	${\frac {d}{dx}}\delta (x-a)\,$
-1	$\delta (x-a)\,$
0	$H(x-a)\,$
1	$(x-a)H(x-a)\,$
2	${\frac {(x-a)^{2}}{2}}H(x-a)\,$
$\geq 0$	${\frac {(x-a)^{n}}{n+1}}\,H(x-a)\,$

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
$H(x)$ is the Heaviside step function: H(x)=0 for x<0 and H(x)=1 for x>0. The value of H(0) will depend upon the particular convention chosen for the Heaviside step function. Note that this will only be an issue for n=0 since the functions contain a multiplicative factor of x-a for n>0.
$\langle x-a\rangle ^{1}$ is also called the Ramp function.

Integration

Integrating $\langle x-a\rangle ^{n}$ can be done in a convenient way in which the constant of integration is automatically included so the result will be 0 at x=a.

$\int \langle x-a\rangle ^{n}dx={\begin{cases}\langle x-a\rangle ^{n+1},&n\leq 0\\{\frac {\langle x-a\rangle ^{n+1}}{n+1}},&n\geq 0\end{cases}}$

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:

$w=-3N\langle x-0\rangle ^{-1}\ +\ 6Nm^{-1}\langle x-2m\rangle ^{0}\ -\ 9N\langle x-4m\rangle ^{-1}\,$

Shear force:

$S=\int wdx$

$S=-3N\langle x-0\rangle ^{0}\ +\ 6Nm^{-1}\langle x-2m\rangle ^{1}\ -\ 9N\langle x-4m\rangle ^{0}\,$

Bending moment:

$M=-\int Sdx$

$M=3N\langle x-0\rangle ^{1}\ -\ 3Nm^{-1}\langle x-2m\rangle ^{2}\ +\ 9N\langle x-4m\rangle ^{1}\,$

Slope:

$u'={\frac {1}{EI}}\int Mdx$

Because the slope is not zero at x=0, a constant of integration, c, is added

$u'={\frac {1}{EI}}\left({\frac {3}{2}}N\langle x-0\rangle ^{2}\ -\ 1Nm^{-1}\langle x-2m\rangle ^{3}\ +\ {\frac {9}{2}}N\langle x-4m\rangle ^{2}\ +\ c\right)\,$

Deflection:

$u=\int u'dx$

$u={\frac {1}{EI}}\left({\frac {1}{2}}N\langle x-0\rangle ^{3}\ -\ {\frac {1}{4}}Nm^{-1}\langle x-2m\rangle ^{4}\ +\ {\frac {3}{2}}N\langle x-4m\rangle ^{3}\ +\ cx\right)\,$

The boundary condition u=0 at x=4m allows us to solve for c=-7Nm²

See also

• Macaulay brackets

External Links

• Singularity Functions (Tim Lahey)
• Singularity functions (J. Lubliner, Department of Civil and Environmental Engineering)
• Beams: Deformation by Singularity Functions (Dr. Ibrahim A. Assakkaf)