Singularity function

Integrating ${\displaystyle \langle x-a\rangle ^{n}}$ 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.

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

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 downward forces and sagging bending moments being positive.

Load distribution:

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

Shear force:

${\displaystyle S=\int w\,dx}$

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

Bending moment:

${\displaystyle M=-\int S\,dx}$

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

Slope:

${\displaystyle u'={\frac {1}{EI}}\int M\,dx}$

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

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

Deflection:

${\displaystyle u=\int u'\,dx}$

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

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

• Macaulay brackets
• Macaulay's method

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