Question $3$

A circular beam, which is built into a wall, is subjected to an axial force $P$ and a torque $T$ as shown in

Figure $3.$

Figure $3$: Circular beam with axial force and torque

$($i$)$ Consider the face where the shaft is built into the wall at B$.$ Make a sketch of an infinitessimal

element of material at B$,$ at beam's outer diameter at the $12$ o'clock position, with one face

parallel to the wall. Clearly indicate the direction of any non$-$zero stresses acting on this element.

Is this element in a state of plane stress? If so$,$ what is the plane? $($Make a clearly labelled sketch

showing your choice of axes$).$

$($ii$)$ Let $P=120kN$ and $T=2kNm.$ Calculate the stresses due to $P$ and $T$ at $B,$ and hence

calculate the maximum shear stress at $B.$