EXAMPLE $3$

A compression bar having a square cross section of width $b$ must support a load $P=35$ kN $($See figure$).$ The bar is constructed from two pieces of material that we connected by a glued joint along plane pq$,$ which is at an angle $=40$ to the vertical. The material is a structural plastic for which the allowable stresses in compression and shear are $7\mathrm{.}5$ MPa and $4\mathrm{.}0$ MPa, respectively. Also the allowable stresses in the glued joint are $5\mathrm{.}2$ MPa in compression and $3\mathrm{.}4$ MPa in shear.

Determine the minimum width $b$ of the bar.