Problem $3($Adopted from Final Exam $-$ Fall $2023)$

A horizontal pipe that is aligned along the $x-$axis is being tightened using a pipe wrench as shown in the picture.

a$)$ Draw a free body diagram of this pipe with all external forces acting on it and find the reactions on the horizontal pipe at $x=0$ in$.$

b$)$ Find the minimum diameter $D$ of the rod considering ${}_{\text{A}\text{L}\text{L}\text{O}\text{W}}=24$ksi.

c$)$ Find the minimum diameter $D$ of the rod considering ${}_{\text{A}\text{L}\text{L}\text{O}\text{W}}=18$ksi.

d$)$ What is the minimum diameter $D$ of the rod?

e$)$ Draw the Mohr's circles for points $F$ and $H$ indicated on the cross$-$section $($Section $a-a)$ and sketch their corresponding states of stress using $3$D stress blocks.

Note:

For a semi$-$circle:

For a full circle: $I=\frac{r^4}{4},J=\frac{r^4}{2}$