Question $3(20$ marks$)$

Figure $3$a shows a pipe of outer radius $R$ and thickness $t$ fixed at the bottom end. The origin of the $xyz-$coordinate system is at the centre $($D$)$ of the fixed end. Two horizontal forces are applied at the centre $($B$)$ of the free end, i$.$e$.P$ in the negative $x-$direction and $F$ in the positive $y-$direction.

$($a$)$ Determine the reaction forces and moments $($including torsion if any$)$ at the fixed end in terms of $P,F$ and other necessary parameters. Present your results on the cross section of the fixed end in a figure similar to Figure $3$b $($which shows part of the pipe viewing from the bottom$).$ Indicate forces by single$-$headed arrows and moments by doubleheaded arrows.

$(12$ marks$)$

$($b$)$ Consider a material point C with coordinates $(0,R,0).$ Explain whether and how each of the force and moment components obtained above produces stress components at point $C.$ Show the directions $($by arrows$)$ of only the non$-$zero stress components on the visible faces of the cube element similar to Figure $3$c$.$ No calculation or algebraic expression is needed for the stress magnitude.

$(8$ marks$)$

Figure $3$c $($Cube element for point C$)$