Gere & Goodno — Torsion: Problem 3.8-18

3.8-18 Two pipes (\(L_1 = 2.5\text{ m}\) and \(L_2 = 1.5\text{ m}\)) are joined at B by flange plates (thickness \(t_f = 14\text{ mm}\)) with five bolts (\(d_{bf} = 13\text{ mm}\)) arranged in a circular pattern (see figure). Also, each pipe segment is attached to a wall (at A and C, see figure) using a base plate (\(t_b = 15\text{ mm}\)) and four bolts (\(d_{bb} = 16\text{ mm}\)). All bolts are tightened until just snug. Assume \(E_1 = 110\text{ GPa}\), \(E_2 = 73\text{ GPa}\), \(\nu_1 = 0.33\), and \(\nu_2 = 0.25\). Neglect the self-weight of the pipes, and assume the pipes are in a stress-free state initially. The cross-sectional areas of the pipes are \(A_1 = 1500\text{ mm}^2\) and \(A_2 = (3/5)A_1\). The outer diameter of Pipe 1 is 60 mm. The outer diameter of Pipe 2 is equal to the inner diameter of Pipe 1. The bolt radius \(r = 64\text{ mm}\) for both base and flange plates. (a) If torque \(T\) is applied at \(x = L_1\), find an expression for reactive torques \(R_1\) and \(R_2\) in terms of \(T\). (b) Find the maximum load variable \(T\) (i.e., \(T_{max}\)) if allowable torsional stress in the two pipes is \(\tau_{allow} = 65\text{ MPa}\). (c) Draw torsional moment (TMD) and torsional displacement (TDD) diagrams. Label all key ordinates. What is \(\phi_{max}\)? (d) Find \(T_{max}\) if allowable shear and bearing stresses in the base plate and flange bolts cannot be exceeded. Assume allowable stresses in shear and bearing for all bolts are \(\tau_{allow} = 45\text{ MPa}\) and \(\sigma_{allow} = 90\text{ MPa}\). (e) Remove torque \(T\) at \(x = L_1\). Now assume the flange-plate bolt holes are misaligned by some angle \(\beta\) (see figure). Find the expressions for reactive torques \(R_1\) and \(R_2\) if the pipes are twisted to align the flange-plate bolt holes, bolts are then inserted, and the pipes released. (f) What is the maximum permissible misalignment angle \(\beta_{max}\) if allowable stresses in shear and bearing for all bolts [from part (d)] are not to be exceeded?