Problem $3\mathrm{.}46$:

A cantilevered beam is loaded as shown. The cross section at the wall is shown, with points of interest $A($at the top$),B($at

the center$),$ and $C($midway between A and $B).$

$($a$)$ Sketch the stress elements at points $A,B,$ and $C.$ Calculate the magnitudes of the stresses acting on the stress

elements. Do not neglect transverse shear stress.

$($b$)$ Find the maximum shear stresses $($i$.$e$.,$ from Mohr's circle$)$ for the stress elements at $A,B,$ and $C.$

$($c$)$ Assuming the critical stress element is the one with the largest maximum shear stress from part $($b$),$ which point is

critical?

$($d$)$ If the length of the beam is varied from longer to shorter, the critical stress element will switch from point $A$ to

point $B.$ At what length does this happen?