22-Mec-B1 Advanced Machine Design · May 2015
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Reference texts. R. G. Budynas & J. K. Nisbett, Shigley’s Mechanical Engineering Design (10th ed.) — deflection §4, fatigue §6, shafts §7, bolted joints §8, journal bearings §12, brakes & clutches §16; R. C. Juvinall & K. M. Marshek, Fundamentals of Machine Component Design; R. L. Norton, Machine Design: An Integrated Approach; R. C. Hibbeler, Mechanics of Materials.
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) The 0.2 % offset proof (yield) strength, $\sigma_{0.2}$. Many engineering metals—aluminium alloys, copper alloys and most heat-treated steels—show no sharp yield point, so the elastic-to-plastic transition cannot be read directly from the stress–strain curve. The proof strength is defined instead by the offset method: a line parallel to the initial elastic slope is drawn from a strain of $0.002$ (that is, $0.2\%$), and its intersection with the curve fixes $\sigma_{0.2}$. Physically it is the stress that leaves a permanent plastic strain of $0.2\%$ after the load is removed; it is the value quoted as the “yield strength” $S_y$ in design tables for such materials.
(b) Hollow versus solid shaft. In both bending and torsion the stress varies linearly with radius—$\tau=Tr/J$ and $\sigma=Mc/I$—so the material near the axis is lightly stressed and contributes little strength while carrying its full weight. Removing that low-stress core gives a hollow shaft a markedly higher strength-to-weight and stiffness-to-weight ratio: for the same outside diameter the polar moment $J$ is barely reduced, yet a large mass is saved, which also raises the critical (whirling) speed for a given mass. The disadvantages are practical: hollow shafts cost more to manufacture (deep boring or tube forming), are harder to machine keyways, splines and shoulders into, have thin walls that are prone to local buckling or crippling under torsion and to denting, and are more sensitive to stress concentrations and to eccentricity between bore and outside diameter.
(c) Effect of mean stress on fatigue. Fatigue life depends not only on the alternating stress amplitude $\sigma_a$ but also on the steady (mean) stress $\sigma_m$ superimposed on it. A tensile mean stress is detrimental—it opens micro-cracks and reduces the alternating stress the part can survive for a given life, so the safe operating amplitude falls as $\sigma_m$ rises, reaching zero when $\sigma_m$ approaches the ultimate (or yield) strength. This trade-off is captured by the failure lines drawn in the $\sigma_a$–$\sigma_m$ plane:
Conversely a compressive mean stress is beneficial—it tends to close cracks and lengthens life, which is exactly why surface treatments such as shot peening and case hardening (which leave a compressive residual surface stress) improve fatigue performance.
(d) Eccentricity ratio $\varepsilon=1$ in a journal bearing. The eccentricity ratio is $\varepsilon=e/c_r$, where $e$ is the offset of the journal centre from the bearing centre and $c_r$ is the radial clearance. The minimum film thickness is $h_0=c_r(1-\varepsilon)$. When $\varepsilon=1$ the journal has moved a full radial clearance and $h_0=c_r(1-1)=0$: the oil film has vanished and there is metal-to-metal contact between journal and bearing. Hydrodynamic (full-film) lubrication has broken down into boundary/mixed lubrication, giving high friction, rapid wear, local heating and—if sustained—seizure. It represents the limiting, failed condition; a healthy bearing runs at $\varepsilon$ well below one (typically $0.6$–$0.8$).
