Consider a beam having open cross$-$section, $90-$deg L shape cross$-$section $($shown below$).$ The beam is made from titanium grade $5.$ The wall thickness, $\backslash ($ t $\backslash ),$ is uniform and much smaller than the flange distance, $\backslash ($ a $\backslash ).$

The centroid is located at: $\backslash (\backslash$frac$\{\backslash$sqrt$\{2\}\}\{4\}$ a $\backslash )$ to the right of the common corner.

a$)$ Determine the moments of inertia of this $\backslash (\backslash$mathrm$\{$c$\}/\backslash$mathrm$\{$s$\}\backslash ).(20$ pts$)$

b$)$ Determine the shear flow distribution over the cross section associated with the shear force $\backslash ($ S$\_\{$y$\}\backslash ),$ and sketch $($draw$)$ the result. Please start your calculation at the top of the flange. $(30$ pts $)$

c$)$ Demonstrate that the force created by the shear flow in the vertical direction is equal to $\backslash ($ S$\_\{\backslash$mu$\}\backslash )(15$ pts$)$

d$)$ What is the maximum shear stress on the cross section associated with the transverse shear load, $\backslash ($ S$\_\{$y$\}\backslash )?$ At what point is it found? $(10$ pts$)$

e$)$ Determine the location of the shear center. Explain physically using equations, why the shear center is in such location. $(15$ pts$)$

f$)$ If an additional torque, $\backslash ($ T $\backslash ),$ is applied to the structure at the shear center, what is the shear stress in the structure and where is it maximum? $(10$ pts $)$