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 Vz 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 $($ Iyy and Izz $)$ and the product of inertia $($ Iyz $)$ 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 Vz passes through the shear center of the open cross section, find the shear

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

flow from stringer i to stringer j$).$ Express the results in terms of Vz$.$

d$)$ Find the value of e for which Vz passes through the shear center of the open section.