Representative Problems

$4\mathrm{.}5-10$ A beam ABC is simply supported at A and $B$ and has an overhang $BC($see figure$).$ The beam is loaded by two forces $P$ and a clockwise couple of moment $Pa$ at $D$ that act through the arrangement shown.

$($a$)$ Draw the shear$-$force and bending$-$moment diagrams for beam ABC.

$($b$)$ If moment $Pa$ at $D$ is replaced by moment $M,$ find an expression for $M$ in terms of variables $P$ and $a$ so that the reaction at $B$ goes to zero. Plot the associated shear$-$force and bending$-$moment diagrams for beam ABC.