Singularity Functions

The loading of beams can be determined from a superposition of singularity functions for the load distribution function $q(x)$.

Singularity functions are defined such that:

$$f(x)=⟨x-a{⟩}^n=\{\begin{array}{ll}(x-a{)}^n& x\ge a,\\ 0& x<a\end{array}$$

Essentially, the function is turned “on” if $x\ge a$ and turned off otherwise.

They can be integrated as:

$${\int }_{-\infty }^x⟨x-a{⟩}^{n}dx=\{\begin{array}{ll}⟨x-a{⟩}^{-1}& n=-2\\ ⟨x-a{⟩}^0& n=-1\\ \frac{⟨x-a{⟩}^{n+1}}{n+1}& n\ge 0\end{array}$$

The functions are undefined for values of $x=a$.

Beam Loading Example

We have a beam loaded like so:

The loading can be written as:

$$q=R_1⟨x{⟩}^{-1}-200⟨x-4{⟩}^{-1}-100⟨x-10{⟩}^{-1}+R_2⟨x-20{⟩}^{-1}$$

The shear can then be found by integrating once:

$$V=\int qdx=R_1⟨x{⟩}^0-200⟨x-4{⟩}^0-100⟨x-10{⟩}^0+R_2⟨x-20{⟩}^0$$

The moment can then be found by integrating again:

$$M=\int Vdx=R_1⟨x{⟩}^1-200⟨x-4{⟩}^1-100⟨x-10{⟩}^1+R_2⟨x-20{⟩}^1$$

We can find $R_1$ and $R_2$ with:

$$\begin{array}{rl}\sum F_y=0& ⟶R_1-200-100+R_2=0\\ \sum M_{R_2}=0& ⟶-R_1(20)+200(16)+100(10)=0\\ & \therefore R_1=210,R_2=90\end{array}$$