Problem $\backslash$#$1$: $(35$ points$)$ A closed stringer$-$web cross$-$section with three stringers and three webs is shown in Figure $1.$ The area of stringer $\backslash$#$1$ is $\backslash ($ A $\backslash ),$ the area of stringer $\backslash (\backslash$# $2\backslash )$ is $\backslash (2$ A $\backslash ),$ and the area of stringer $\backslash$#$3$ is $\backslash (3$ A $\backslash ).$ The overall dimensions of the cross$-$section are given on the figure in terms a length $\backslash ($ h $\backslash ).$ The $\backslash ($ y$-$z $\backslash )$ coordinate system has its origin at the centroid $\backslash ($ C $\backslash )$ of the cross section and is oriented as shown. $($Note that the centroid $\backslash ($ C $\backslash )$ is not located to scale on the figure.$)$ The crosssection shown in the figure is subjected to a bending moment $\backslash ($ M$\_\{$y$\}\backslash )$ in the negative $\backslash ($ y $\backslash )-$direction, as shown.

$[$Note: do the calculations in parts a$)$ and b$)$ below very carefully because they will affect the results of parts c$),$ d$),$ and e$).]$

a$)$ Determine the location of the centroid $\backslash ($ C $\backslash )$ of the cross$-$section. Specify the location as horizontal and vertical distances from stringer $\backslash$#$1.$ Express the results in terms of the length h$.$

b$)$ Determine the moments of inertia $(\backslash ($ I$\_\{$y y$\}\backslash )$ and $\backslash ($ I$\_\{$z z$\}\backslash ))$ and the product of inertia $(\backslash ($ I$\_\{$y z$\}\backslash ))$ of the crosssection. Express the results in terms of the length $\backslash ($ h $\backslash )$ and the area $\backslash ($ A $\backslash ).$

c$)$ Calculate the bending stresses $\backslash (\backslash$sigma$\_\{$x x$\}\backslash )$ acting on stringers $\backslash$#$1,\backslash$#$2,$ and $\backslash$#$3.$ Label them $\backslash (\backslash$sigma$\_\{1\},\backslash$sigma$\_\{2\}\backslash ),$ and $\backslash (\backslash$sigma$\_\{3\}\backslash )$ respectively. Express the results in terms of the bending moment $\backslash ($ M$\_\{$y$\}\backslash ),$ the length $\backslash ($ h $\backslash ),$ and the area $\backslash ($ A $\backslash ).$

d$)$ Verify that the bending stresses obtained in part a$)$ exert the expected bending moments about the $\backslash ($ y $\backslash )-$ and $\backslash ($ z $\backslash )-$axes, respectively.

e$)$ Determine the neutral axis of the cross section. Draw a clear picture describing its orientation relative to the $\backslash ($ y $\backslash )-$axis. Figure $1$