Question 1 of 6: Short-answer concepts

Question 1: Short-answer concepts (10 marks)

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) Five green (design-for-environment) criteria. A machine element earns green-design credit when it is designed for: (1) material efficiency — minimum mass and no over-specification of alloy content; (2) energy efficiency in manufacture and in service (low friction, low idle loss); (3) recyclability / design-for-disassembly — separable, labelled, mono-material parts joined without permanent bonds; (4) durability and reparability — long service life, replaceable wear parts, standard fasteners; (5) use of renewable / low-toxicity / recycled materials and avoidance of hazardous substances (e.g. hexavalent-chromium plating). Reduced packaging, remanufacturability and a documented life-cycle assessment are equally acceptable answers.

(b) Hollow vs. solid shaft. Bending and torsional stiffness scale with the second moment of area, which grows as the fourth power of diameter, so material near the axis contributes almost nothing to strength: $$I=\frac{\pi}{64}\left(d_o^{4}-d_i^{4}\right),\qquad J=\frac{\pi}{32}\left(d_o^{4}-d_i^{4}\right).$$ Removing the lightly-stressed core gives a much higher stiffness-to-weight and strength-to-weight ratio, and the larger outer diameter raises the critical whirling speed for the same mass. Disadvantages: higher manufacturing cost (deep boring or forming over a mandrel); thin walls are prone to local buckling and are harder to key, press-fit or machine to a bore tolerance; and for a given outside diameter the hollow shaft is weaker than the solid one, so the weight saving is only realised when the outer diameter is allowed to grow.

(c) Plane-strain tension carries more load before yielding. In uniaxial tension only one principal stress acts, so von Mises yielding begins when that stress reaches $S_y$. In plane strain the specimen is prevented from contracting in one transverse direction, which generates a second (constraining) tensile stress. Elastically $\sigma_2=\nu\,\sigma_1$ under the plane-strain constraint, and at the onset of plastic flow $\nu\to\tfrac12$, giving $\sigma_2=\tfrac12\sigma_1$. Substituting into the von Mises criterion, $$\sigma_1^{2}-\sigma_1\sigma_2+\sigma_2^{2}=S_y^{2}\ \Rightarrow\ \sigma_1\sqrt{1-\tfrac12+\tfrac14}=S_y,$$ so the axial stress at yield rises to $$\sigma_1=\frac{S_y}{\sqrt{3}/2}=\frac{2}{\sqrt{3}}\,S_y\approx 1.155\,S_y.$$ The hydrostatic (mean) stress does no work in the deviatoric von Mises measure, so the extra transverse tension raises the axial stress needed to reach the same shear-distortion energy — the material yields at a ≈15.5 % higher axial load than in uniaxial tension.

(d) Minimum film thickness and viscosity. In a hydrodynamic journal bearing the load-carrying pressure film is generated by the wedge action of the rotating journal dragging lubricant into a converging gap. The governing group is the Sommerfeld (bearing characteristic) number $$S=\left(\frac{r}{c}\right)^{2}\frac{\mu N}{P},$$ where $\mu$ is the absolute viscosity, $N$ the speed and $P$ the projected load. The minimum film thickness is $h_0=c\,(1-\varepsilon)$, and the eccentricity ratio $\varepsilon$ decreases (the journal rides more centrally) as $S$ increases. Since $S$ is directly proportional to $\mu$, a higher lubricant viscosity raises $S$, lowers $\varepsilon$, and therefore increases the minimum oil-film thickness — up to the thermal limit where viscosity heating erodes the benefit.