4. (a) A simply supported beam of hollow rectangular section is to be designed for minimum weight to carry a vertical load F, and an axial load P as shown in the Figure Q2.4 below. 2 72 +/ P x2 18 XI- Cross section of beam Figure Q2.4 The deflection of the beam in the y direction under the self-weight and Fy should not exceed 12.7mm. The beam should not buckle either in the yz or the xz plane under the axial load. Assuming the ends of the beam to be pin ended, formulate the optimization problem using xi (i- 1.2.3.4) as design variables for the following data: Fy = 150kg. P= 20,000kg. 1= 3000mm. E = 200 x 10 Pa. p = 7.86x10kg/m, lower bound on x; and x2 = 4mm, upper bound on x; and x2 = 100mm. (b) Formulate the problem stated in part (a) using xy and x as design variables, assuming the beam to have a solid rectangular cross section and find the soliition to the problem.