(a) A simply supported beam of hollow rectangular section is to be designed for minimum weight to carry a vertical load Fy and an axial load P as shown in Fig. 1.28. The deflection of the beam in the y direction under the self-weight and Fy should not exceed 0.5 in. 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, (b) Formulate the problem stated in part (a) using x1 and X2 as design variables, assuming the beam to have a solid rectangular cross section. Also find the solution of the problem using a graphical technique

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(1)SolutionGiven that\begin{array}{l}i=1,2,3,4 \\F y=300(b \\P=40,000 \mathrm{lb} \\l=120 i n, \\\rho=0.284 \mathrm{lb} / \mathrm{in}^{3}\end{array}(A).Load is symmetrically placed on Span,Reaction on rach Support =\frac{\omega}{2}\begin{array}{c}V_{A}+V_{B}=\frac{\omega}{2} \\S \cdot F=+\frac{\omega}{2} \quad(\text { Section } A C \text { ) } \\S \cdot f=-\frac{\omega}{2} \quad \text { (Section BC) }\end{array}At C, Sf changes from positive to regative,(2)\begin{array}{l} M_{x}=\frac{\omega}{2} x \text { Sagging } \\x=0, M_{x}=0 \\{\left[x=1 / 2, M_{x}=\frac{\omega d}{4}\right] }\end{array} ...