Figure $1$

Problem #$1$: $(40$ 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 #$4.$ The area of stringer #$1$ is $3$A; the area of

each stringer #$2,$ #$3,$ and #$4$ is A$.$

The cross$-$section shown in the figure is subjected to a vertical shear force V$\_($z$)$ in the positive z $-$

direction acting at a distance e to the right of stringers #$2$ and #$4,$ as shown. The overall dimensions

of the cross$-$section are given on the figure. The y$-$z coordinate system has its origin at the centroid

C of the cross section and is oriented as shown. $($The centroid C 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 C of the cross$-$section. Specify the location of C as

horizontal and vertical distances from stringer #$3.$

b$)$ Determine the moments of inertia $($ I$\_($yy$)$ and I$\_($zz$))$ and the product of inertia $($ I$\_($yz$))$ of the cross$-$

section $($in which the y$-$z coordinate system has its origin at the centroid C of the cross

section and is oriented as shown$).$ Express the results in terms of area A$.$

c$)$ Assuming that V$\_($z$)$ passes through the shear center of the open cross section, find the shear

flows q$\_(12),$q$\_(23),$ and q$\_(34)$ on the open cross section. $($Use the convention that q$\_($ij$)$ is the shear

flow from stringer i to stringer j $).$ Express the results in terms of V$\_($z$).$

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