1. Show that the strain energy in a bar of length L and cross sectional area A, hanging from a ceiling and subjected to its own weight is given by (at any section, the force acting is the weight of the material below that section) (25 Pts) Ap gL U = 6E 2. Two bars of equal length L and cross-sectional area A are pin-supported and loaded by a force F as shown below. Derive an expression for the vertical displacement at point A, using the direct work-energy method, in terms of L, F, A and the Young's modulus E. (25 Pts) 450 3. Consider a simply supported beam of length L subjected to a uniform load w N/m. Calculate the strain energy due to both flexural stress and shear stress for (a) a rectangular cross-section of depth times height bx h, (b) a circular cross-section with radius r. What is the ratio of the shear- to-flexural strain energies in each case? (25pts) 4. Consider a cantilevered beam of length L and constant cross-section subjected to a uniform load w N/m. The beam is built-in at x = 0 and has a Young's modulus E. Use Castigliano's theorem to calculate the deflection at x = L. Consider only the flexural strain energy. [Hint: place a fictitious "dummy" load F at x = L and set to zero once Castigliano's theorem has been applied] (25 Pts)