Problem $2$:

A composite beam is fabricated by bolting two $\backslash ($ b$=\backslash )3\mathrm{.}8-$in$.-$wide by $\backslash ($ d$=14\backslash )-$in$.-$deep timber planks to the sides of a $\backslash ($ t$=0\mathrm{.}6-\backslash$mathrm$\{$in$\}\backslash ).$ by $\backslash ($ d$=14-\backslash$mathrm$\{$in$\}\backslash ).$ steel plate. The moduli of elasticity of the timber and the steel are $\backslash (1,930\backslash$mathrm$\{$ksi$\}\backslash )$ and $\backslash (30,800\backslash$mathrm$\{$ksi$\}\backslash ),$ respectively. Two concentrated loads $\backslash ($ P $\backslash )$ are applied as shown, and the positions of the supports and loads are given by $\backslash ($ L$\_\{$I$\}=\backslash )8$ ft and $\backslash ($ L$\_\{2\}=18\backslash$mathrm$\{$ft$\}\backslash ).$ Assume that the allowable bending stresses of the timber and the steel are $1,070$ psi and $\backslash (19,800\backslash$mathrm$\{$psi$\}\backslash ),$ respectively.

a$.$ Determine the largest acceptable magnitude for concentrated loads $\backslash ($ P $\backslash ).($You may neglect the weight of the beam in your calculations.$)$

b$.$ Determine the maximum bending stress in the timber beam and the maximum bending stress in the steel plate when the largest acceptable value of the load $\backslash ($ P $\backslash )$ is applied. Show the location along the beam and the location in the cross$-$section of each stress element on the diagrams at right $($or your own$).$ Note that it is acceptable to represent either a maximum positive or a maximum negative value of bending stress, but the location that you report must match the sign of the maximum bending stress that you calculated.

c$.$ For each of these maximum stresses, draw a stress element in the appropriate plane. Include coordinate axes and label the stresses, showing proper direction and magnitude. $($Note that you must determine the appropriate plane on which to draw the stress element $-$ that is$,$ the plane corresponding to the viewing direction that makes the most sense.$)$