고체역학 (Gere) — Axially Loaded Members: Problem 2.4-14

2.4-14 A hollow circular pipe (see figure) supports a load P that is uniformly distributed around a cap plate at the top of the lower pipe. The inner and outer diameters of the upper and lower parts of the pipe are \(d_1 = 50\text{ mm}\), \(d_2 = 60\text{ mm}\), \(d_3 = 57\text{ mm}\), and \(d_4 = 64\text{ mm}\), respectively. Pipe lengths are \(L_1 = 2\text{ m}\) and \(L_2 = 3\text{ m}\). Neglect the self-weight of the pipes. Assume that cap plate thickness is small compared to \(L_1\) and \(L_2\). Let \(E = 110\text{ GPa}\)*. (a) If the tensile stress in the upper part is \(\sigma_1 = 10.5\text{ MPa}\), what is load \(P\)? Also, what are reactions \(R_1\) at the upper support and \(R_2\) at the lower support? What is the stress \(\sigma_2\) (MPa) in the lower part? (b) Find displacement \(\delta\) (mm) at the cap plate. Plot the axial force diagram (AFD) [\(N(x)\)] and axial displacement diagram (ADD) [\(\delta(x)\)]. (c) Add the uniformly distributed load \(q\) along the centroidal axis of pipe segment 2. Find \(q\) (kN/m) so that \(R_2 = 0\). Assume that load \(P\) from part (a) is also applied. *Note: The original text states \(E = 110\text{ MPa}\), which is likely a typo for \(110\text{ GPa}\) common for copper/brass alloys. \(110\text{ GPa}\) is used for realistic results.