$($Maximum $2+3$ points$)$ A prismatic bar of length $L$ hangs from a fixed support. If the origin is located at the centroid of the fixed cross$-$section with the $Z$ axis pointing downwards, show that the stress distribution ${}_{zz}={}_{0}g(L-z),$ other stress components are zero, satisfies the equation of equilibrium and the traction boundary conditions at free surfaces of the bar. Show that the corresponding strains satisfy the equation of compatibility, and hence can be integrated using the boundary conditions $u=v=w=$del$\frac{u}{d}$elz$=$del$\frac{v}{d}$elz$=$del$\frac{u}{d}$ely$=0$ at