Question 1 of 6: Short-Answer Set — Green Design, Shafts, Bearings & Yield Criteria

Question 1: Short-Answer Set — Green Design, Shafts, Bearings & Yield Criteria (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 (sustainable) design criteria. A design is “green” when it reduces environmental burden across the whole life cycle. Five accepted criteria are: (1) select benign materials — recyclable, renewable, non-toxic, low embodied-energy stock; (2) minimise material use — lightweight, dematerialised parts that still meet the load requirement; (3) energy efficiency — low energy in both manufacture and in service; (4) design for disassembly and recyclability — joints that separate cleanly so components are reused, remanufactured or recycled at end of life; and (5) minimise waste, emissions and pollution across manufacture, use and disposal. Design for durability/long life and design for reduced packaging and transport are further valid entries.

(b) Hollow versus solid shaft. In torsion and bending the resisting capacity scales with the polar/second moment of area, which grows as the fourth power of radius, so material at the outer radius is by far the most effective at carrying stress. Material near the axis contributes almost nothing to strength but adds most of the weight. A hollow shaft removes this ineffective core, so for the same weight it offers a larger section modulus and polar moment — hence higher strength-to-weight and stiffness-to-weight ratios and a higher fundamental (whirl) frequency. The disadvantages are: higher manufacturing cost and difficulty (accurate boring, drawn or welded tube, concentricity control); a thin wall becomes prone to local buckling / crippling; keyways and press-fits at the bore are harder to accommodate and raise stress concentrations; and inspection of the bore is more difficult.

(c) Minimum oil film thickness and viscosity. In hydrodynamic (full-film) lubrication the load is carried by a wedge of pressurised oil. The minimum film thickness is $h_0 = c_r(1-\varepsilon)$, where $c_r$ is the radial clearance and $\varepsilon$ the eccentricity ratio. The eccentricity is governed by the dimensionless Sommerfeld number $S = \left(\dfrac{r}{c_r}\right)^2 \dfrac{\mu N}{P}$: a higher viscosity $\mu$ (like higher speed $N$ or lower unit load $P$) raises $S$, which lowers $\varepsilon$ and therefore increases $h_0$. In short, minimum film thickness increases with lubricant viscosity — a thicker, more viscous oil builds a thicker separating film for a given load and speed (subject to the accompanying rise in friction power and temperature).

(d) von Mises versus Tresca conservatism. No. Tresca’s maximum-shear-stress criterion is the more conservative of the two: its yield surface is a hexagon inscribed within the von Mises ellipse, so for any biaxial state Tresca predicts yielding at an equal or lower load. Von Mises permits up to $1/\cos 30^\circ = 0.577/0.5 \approx 1.155$ times more shear stress before yield (the two agree only in uniaxial and in balanced-biaxial tension). Thus von Mises is less conservative — it is also the more accurate for ductile metals.

(e) Higher capacity in plane-strain tension. In uniaxial tension the bar is free to contract laterally, so the stress state is one-dimensional and yielding begins when the axial stress reaches $S_y$. In plane-strain tension one transverse direction is constrained ($\varepsilon = 0$), which raises a transverse (constraint) stress in that direction. This constraint stress increases the hydrostatic (mean) component of the stress state without adding to the deviatoric intensity that actually drives yield. Because yielding depends only on the deviatoric part, a higher applied axial stress is now needed to reach the criterion. Working the von Mises criterion for the plane-strain state gives the raised axial yield

$$\sigma_{y,\text{plane strain}} = \frac{2}{\sqrt{3}}\,S_y \approx 1.155\,S_y,$$

so the material carries about 15.5 % more axial load before yielding than in uniaxial tension — the origin of the “plastic constraint” strengthening seen ahead of notches and in thick sections.