Question $6$: The simple beam shown in the figure supports a concentrated load $P$ acting at distance a from the left$-$hand support and distance $b$ from the right$-$hand support. Using Castigliano's theorem, determine the deflection $D$ at point $D$ in millimeters where the load is applied. Assume $P=45kN,a=4m,b=2m,L=6m,E=210$ GPa $,$ and $I=5\mathrm{.}5{10}^{-4}{m}^4.$

Hint: Develop the formula for ${}_D$ using the symbolic notation first and calculate the value later.