$2\mathrm{.}7-1$ A prismatic bar $AD$ of length $L,$ cross$-$sectional area $A,$ and modulus of elasticity $E$ is subjected to loads $5P,3P,$ and $P$ acting at points $B,C,$ and $D,$ respectively $($see figure$).$ Segments $AB,BC,$ and $CD$ have lengths $\frac{L}{6},\frac{L}{2},$ and $\frac{L}{3},$ respectively.

$($a$)$ Obtain a formula for the strain energy $U$ of the bar.

$($b$)$ Calculate the strain energy if $P=27kN,L=130cm,A=18c{m}^2,$ and the material is aluminum with $72$ GPa $.$

PROB. $2\mathrm{.}7-1$