Chapter 3.8, Problem 16P

1. Suppose that a beam is fixed at its ends $x=0\text{and}x=L$. Show that its shape is given by $y\left(x\right)=\frac{\omega }{24EI}\left(x^4-2L{x}^3+L^2{x}^2\right)$.
2. Show that the roots of $y\text{'}\left(x\right)=0\text{are}x=0,x=L,\text{and}x=L/2$, so it follows (why?) that the maximum deflection of the beam is $y_{\max }=y\left(\frac{L}{2}\right)=\frac{\omega L^4}{384EI}$, one-fifth that of a beam with simply supported ends.