Problem $\backslash$#$1$: $(40$ points$)$ An open stringer$-$web cross$-$section is shown in Figure $1.$ Web $2-3$ is a quarter circle $($of radius $1\mathrm{.}2$ m $)$ centered at stringer $\backslash$#$4.$ The area of stringer $\backslash$#$1$ is $3$ A ; the area of each stringer $\backslash$#$2,\backslash$#$3,$ and $\backslash$#$4$ is $\backslash ($ A $\backslash ).$

The cross$-$section shown in the figure is subjected to a vertical shear force $\backslash ($ V$\_\{$z$\}\backslash )$ in the positive $\backslash ($ z $\backslash )$ direction acting at a distance $\backslash ($ e $\backslash )$ to the right of stringers $\backslash$#$2$ and $\backslash$#$4,$ as shown. The overall dimensions of the cross$-$section are given on the figure. 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. $($The centroid $\backslash ($ C $\backslash )$ is not located to scale on the figure.$)$

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

a$)$ Determine the location of the centroid $\backslash ($ C $\backslash )$ of the cross$-$section. Specify the location of $\backslash ($ C $\backslash )$ as horizontal and vertical distances from stringer $\backslash$#$3.$

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 $($in which 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$).$ Express the results in terms of area $\backslash ($ A $\backslash ).$

c$)$ Assuming that $\backslash ($ V$\_\{$z$\}\backslash )$ passes through the shear center of the open cross section, find the shear flows $\backslash ($ q$\_\{12\},$ q$\_\{23\}\backslash ),$ and $\backslash ($ q$\_\{34\}\backslash )$ on the open cross section. $($Use the convention that $\backslash ($ q$\_\{\backslash$mathrm$\{$ij$\}\}\backslash )$ is the shear flow from stringer $\backslash ($ i $\backslash )$ to stringer $\backslash ($ j $\backslash )).$ Express the results in terms of $\backslash ($ V$\_\{$z$\}\backslash ).$

d$)$ Find the value of $\backslash ($ e $\backslash )$ for which $\backslash ($ V$\_\{$z$\}\backslash )$ passes through the shear center of the open section.