Problem #$4$: $(40$ points$)$ Throughout this problem, the $y-z$ coordinate system is as described in Problem #$2.$ Each stringer has an area $A=\mathrm{.}002m^2,$ and the length $h=1\mathrm{.}2m,$ as in Problem #$3.$ The shear modulus is a constant $G$ and the web thickness is a constant $t$ everywhere on the cross section.

a$)$ Consider an open stringer$-$web cross$-$section that is formed by taking an artificial cut through stringers #$3$ and #$4$ in Figure $2.$ The open cross$-$section is subjected to a vertical shear force $V_z($not shown in the figure$)$ acting in the positive $z-$direction at a distance $e$ to the left of stringers #$1$ and #$4.$

i$.$ Assuming that $V_z$ passes through the shear center of the open cross section, find the shear flows $q_{12}^{\text{'}},q_{23}^{\text{'}},$ and $q_{41}^{\text{'}}$ on the open cross section. $($Use the convention that $q_{ij}^{\text{'}}$ is the shear flow from stringer $i$ to stringer $j).$ Express the results in terms of $V_z.$

ii$.$ Find the value of $e$ for which $V_z$ passes through the shear center of the open section.

b$)$ Now consider the closed, single$-$celled stringer$-$web cross$-$section shown in Figure $2.$ This closed cross$-$section is subjected to a vertical shear force $V_z($not shown in the figure$)$ acting in the positive $z-$direction applied at stringer #$2.$

i$.$ Using the results in part a$.i),$ find the shear flows $q_{12},q_{23},q_{34},$ and $q_{41}$ on the closed cross section in Figure $2.($Use the convention that $q_{ij}$ is the shear flow from stringer $i$ to stringer $j).$ Express the results in terms of $V_z.$

ii$.$ Find the twist$/$length $$ of the closed cross section in terms of $V_z,G,$ and $t.$