Gere & Goodno — Axially Loaded Members: Problem 2.4-7

2.4-7 A circular bar \(ACB\) of a diameter \(d\) having a cylindrical hole of length \(x\) and diameter \(d/2\) from \(A\) to \(C\) is held between rigid supports at \(A\) and \(B\). A load \(P\) acts at \(L/2\) from ends \(A\) and \(B\). Assume \(E\) is constant. (a) Obtain formulas for the reactions \(R_A\) and \(R_B\) at supports \(A\) and \(B\), respectively, due to the load \(P\) (see figure part a). (b) Obtain a formula for the displacement \(\delta\) at the point of load application (see figure part a). (c) For what value of \(x\) is \(R_B = (6/5)R_A\)? (See figure part a.) (d) Repeat part (a) if the bar is now rotated to a vertical position, load \(P\) is removed, and the bar is hanging under its own weight (assume mass density \(= \rho\)). (See figure part b.) Assume that \(x = L/2\).