Problem #$3$: $(40$ points$)$ The thin$-$walled cross section in Figure $3$ is subjected to a counterclockwise torque $T=20kN-m($not shown on the figure$).$ The wall thickness is $t=10mm$ everywhere on the cross section. The left cell $($#$1)$ is a semicircle of radius $8t$ and a straight edge of length $16t$; and the right cell $($#$2)$ is a combined semicircle of radius $8t$ and three straight edges of length $16t$ each. The shear modulus is $G=25$GPa everywhere on the cross section.

a$)$ Find the algebraic equations that determine the shear flows everywhere on the cross section. $($You don't need to solve them, yet.$)$

b$)$ Using the equations found in part a$),$ determine the shear flows everywhere on the cross section. Draw a clear picture describing where each acts.

c$)$ Find the angle of twist per unit length $$ and the torsional rigidity T$.$R$.$ of the cross section.

Figure $3$