Gere & Goodno — Axially Loaded Members: Problem 2.5-28

2.5-28 Consider the sleeve made from two copper tubes joined by tin-lead solder over distance \(s\). The sleeve has brass caps at both ends that are held in place by a steel bolt and washer with the nut turned just snug at the outset. Then, two “loadings” are applied: \(n = 1/2\) turn applied to the nut; at the same time, the internal temperature is raised by \(\Delta T = 30^\circ\text{C}\). (a) Find the forces in the sleeve and bolt, \(P_S\) and \(P_B\), due to both the prestress in the bolt and the temperature increase. For copper, use \(E_c = 120\text{ GPa}\) and \(\alpha_c = 17 \times 10^{-6}/^\circ\text{C}\); for steel, use \(E_s = 200\text{ GPa}\) and \(\alpha_s = 12 \times 10^{-6}/^\circ\text{C}\). The pitch of the bolt threads is \(p = 1.0\text{ mm}\). Assume \(s = 26\text{ mm}\) and bolt diameter \(d_B = 5\text{ mm}\). (b) Find the required length of the solder joint, \(s\), if shear stress in the sweated joint cannot exceed the allowable shear stress \(\tau_{aj} = 18.5\text{ MPa}\). (c) What is the final elongation of the entire assemblage due to both temperature change \(\Delta T\) and the initial prestress in the bolt? Properties and dimensions from diagram: Upper tube: \(L_1 = 40\text{ mm}\), \(d_1 = 25\text{ mm}\), \(t_1 = 4\text{ mm}\) Lower tube: \(L_2 = 50\text{ mm}\), \(d_2 = 17\text{ mm}\), \(t_2 = 3\text{ mm}\)