25-Nav-B1 Applied Thermodynamics and Heat Transfer (25-Mec-A1) · December 2013
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
National Exams — December 2013 · 98-Mar-B1 Advanced Machine Design. Open-book, 3 hours, 100 marks. The paper requires all of Part I (Problems 1 and 2) plus any three of the four Part II problems (3–6). All six problems are solved in full below.
Reference texts. R. G. Budynas & J. K. Nisbett, Shigley’s Mechanical Engineering Design, 10th ed. (shafts & critical speed §7; bolted joints §8; lubrication & journal bearings §12; brakes §16); R. C. Juvinall & K. M. Marshek, Fundamentals of Machine Component Design (bearings, brakes, impact); R. C. Hibbeler, Mechanics of Materials (bending, impact loading).
Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.
(a) Von Mises vs. Tresca conservatism. No — von Mises is the less conservative of the two. Plotted in principal-stress space, the Tresca (maximum-shear) yield locus is a hexagon inscribed inside the von Mises (distortion-energy) ellipse. The two coincide only along uniaxial tension/compression and balanced biaxial tension; everywhere else the Tresca surface lies inside, so for a given stress state Tresca reaches its limit at a lower load and predicts yielding first. The gap is largest in pure shear, where von Mises permits a shear yield strength of $S_{sy}=0.577\,S_y$ against Tresca’s $0.5\,S_y$ — von Mises allows about 15% more shear stress. Because it fails earlier, Tresca is the more conservative (safer, but heavier) criterion; von Mises matches ductile-metal test data more closely and is the usual design choice.
(b) Hollow versus solid shaft. In bending and torsion the second moment of area and polar moment scale with the fourth power of radius, so the material near the neutral axis carries almost no stress and contributes little strength while adding weight. Removing that low-stress core gives a much higher stiffness-to-weight and strength-to-weight ratio: for the same torque or bending capacity a hollow shaft is significantly lighter, which also raises its critical (whirling) speed and lowers rotating inertia. The bore can additionally route coolant, oil or wiring. The disadvantages are practical: hollow shafts cost more to manufacture (deep boring or tube forming), are prone to local wall buckling/crippling if the wall is made too thin, develop stress concentrations and residual stresses at the bore, are harder to hold to tight concentricity, and complicate the machining of keyways and press-fit seats.
(c) Plane-strain tension. In uniaxial tension the two transverse directions contract freely, so only one principal stress is non-zero. In plane strain one transverse contraction is prevented ($\varepsilon=0$), and that constraint develops an intermediate principal stress; at the fully plastic condition (Poisson’s ratio $\nu\to\tfrac12$) this intermediate stress equals half the axial stress, $\sigma_2=\tfrac12\sigma_1$, with $\sigma_3=0$. Substituting into the von Mises effective stress, $$\sigma' = \sqrt{\sigma_1^{2}-\sigma_1\sigma_2+\sigma_2^{2}} = \sqrt{\sigma_1^{2}-\tfrac12\sigma_1^{2}+\tfrac14\sigma_1^{2}} = \tfrac{\sqrt3}{2}\,\sigma_1 .$$ Yielding ($\sigma'=S_y$) therefore needs $\sigma_1 = \tfrac{2}{\sqrt3}S_y \approx 1.155\,S_y$. The lateral constraint raises the deviatoric-stress threshold, so the material carries roughly 15% more axial stress before yielding than in the unconstrained uniaxial case.
(d) Eccentricity ratio equal to one. The eccentricity ratio is $\varepsilon = e/c_r$, and the minimum oil-film thickness is $h_0 = c_r(1-\varepsilon)$. When $\varepsilon = 1$ the journal has moved through the full radial clearance, so $\boxed{h_0 = c_r(1-1) = 0}$: the journal surface touches the bearing. There is no converging film to generate hydrodynamic pressure, the bearing operates in boundary/metal-to-metal contact, and rapid wear or seizure follows. It represents the limiting, fully-loaded (or start-up) condition and must be avoided in a hydrodynamic design, which always keeps $\varepsilon$ well below unity.
