Problem $1$: Given the steel beam below with the cross$-$section shown. Note the beam has a centroid location $100$ mm from the bottom of the beam in the figure below $60$mm from the top of the beam$).$ The distributed load w$=30\mathrm{.}0$ kn$/$m$.$ The beam has moment of inertia I$=21\mathrm{.}7610^\{6\}$ mm$^\{4\}$ and area A$=9600$ mm$^2.$ The beam is AISI $1006$ HR steel $($see Table A$-20).$ Section A$-$A is $1$ meter from end B$.$

St $=300$ Sy$=170$

Find: a$)$ The factor of safety versus static failure using an appropriate failure theory at section A$-$A and the junction between the top flange and web of the beam $($i$.$e$.,40$ mm from the top edge of the beam$).$

b$)$ The minimum factor of safety versus static failure for the beam and where it occurs. Again, use an appropriate failure theory. Also, note why you are looking at the $(3-$dimensional!$)$ location that you are.