A $90$ sign post of elastic modulus E$,$ cross sectional area A and moment of inertia I is loaded at its end by a force P: a$)$ Determine the strain energy in section AB$.(5$ points$)$ b$)$ Determine the strain energy in section BC$.$ Neglect transverse shear. $(5$ points$)$ c$)$ Using Castigliano$$s theorem, determine the vertical deflection at point C$.(5$ points$)$ d$)$ Repeat part b$)$ including the effect of transverse shear, and assuming a circular cross section. Is the transverse shear significant? Justify your answer. $(5$ points$)$