Two pipes (L₁ = 2.5 m and L₂ =1.5 m) are joined at B by flange plates (thickness t_f =14 mm) with five bolts (d_bf =13 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 mm) and four bolts (d_bb =16 mm). All bolts are tightened until just snug. Assume E₁ = 110 GPa, E₂ = 73 GPa, v₁ = 0.33, and v₂ = 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 A₁= 1500 mm² and A₂ = (3/5)A₁. 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 mm for both base and flange plates.

(a) If torque T is applied at x = L₁, find an expression for reactive torques R₁ and R₂ in terms of T.

(b) Find the maximum load variable T (i.e., T_max) if allowable torsional stress in the two pipes is t_allow = 65 MPa.

(c) Draw torsional moment (TMD) and torsional displacement (TDD) diagrams. Label all key ordinates. What is f_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 t_allow = 45 MPa and s_allow = 90 MPa.

(e) Remove torque T at x = L₁. Now assume the flange-plate bolt holes are misaligned by some angle. Find the expressions for reactive torques R₁ and R₂ if the pipes are twisted to align the flange-plate bolt holes, bolts are then inserted, and the pipes are released.

(f) What is the maximum permissible misalignment angle b_max if allowable stresses in shear and bearing for all bolts are not to be exceeded?