Question $2($a: $12$ points, b: $12$ points, c: $12$ points$)$

In an experiment, consider that the simply supported $\backslash (4\backslash$times $6\backslash )$ beam in Figure $1$ is loaded with an increasing force P $,$ which is representative of the LRFD load combination $0\mathrm{.}9$D$+$W$.$ Considering that the beam material is No$.\backslash (1\backslash$& $\backslash )$ Btr grade Hem$-$Fir species group and assuming dry$-$service and normal temperature conditions, and that incising is needed:

a$)$ Determine the adjusted LRFD bending strength $(\backslash (\backslash$mathrm$\{$F$\}\_\{\backslash$text $\{$bn $\}\}\backslash )).$ Consider that lateral$-$torsional buckling is not a concern.

b$)$ Determine the adjusted LRFD shear strength $(\backslash (\backslash$mathrm$\{$F$\}\_\{\backslash$mathrm$\{$vn$\}\}\backslash )).$

c$)$ It is known that bending or shear failure occurs. Calculate the force P at failure and indicate whether bending or shear failure occurs at this force $($Hint: Use the strength values in parts a and $\backslash ($ b $\backslash ),$ equations that relate bending moment to bending stress and shear force to shear stress, and the shear and moment diagrams$).$

Figure $1.$ Simply supported $\backslash (4\backslash$times $6\backslash )$ beam loaded with an increasing force $\backslash ($ P $\backslash )$