22-Mec-B5 Product Design and Development · December 2016

Question 7 of 7: Material Selection for a Kitchen Table and a Computer Case

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Question 7: Material Selection for a Kitchen Table and a Computer Case (15 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 — Three materials for a kitchen table, and their challenges (3 marks)

Solid hardwood (hard maple or white oak). Stiff, strong, repairable by sanding and refinishing, and the material customers value most. The challenges are all consequences of it being a natural, anisotropic, hygroscopic material: it moves across the grain with humidity by up to 0.2 % for each 1 % change in moisture content, so a wide glued top must be allowed to expand or it will cup or split, and the apron fasteners must be slotted. It must be kiln dried to 6–8 % and kept there; knots, grain runout and colour variation cause yield loss; and the finish, not the wood, carries the resistance to water, heat and red wine.

Engineered wood panel (MDF or particleboard, veneered or melamine faced). Dimensionally stable, flat, cheap, available in large formats and perfectly suited to a pre-finished surface. Its challenges are mechanical and environmental: it has no grain to carry a screw, so edge fixings need inserts or cam-and-dowel hardware; a swollen edge from a spill is permanent because the swelling is fibre expansion, not a surface defect; the panel is heavy for its stiffness; the edge must be banded to be presentable; and the urea-formaldehyde binder brings emission compliance obligations.

Tempered glass on a powder-coated steel or aluminium frame. Hygienic, entirely impervious to water and heat, easy to clean, and visually light. The challenges are that glass is brittle and notch sensitive, so every edge must be ground and the panel must be tempered, which fixes the size and all holes before heat treatment — nothing can be machined afterwards; the top is heavy and awkward to handle safely; it needs elastomeric isolators so the frame cannot load it locally; it shows every fingerprint; and thermal toughening leaves a small but real risk of spontaneous fracture from nickel sulphide inclusions.

B — How the material choice affects the final use (3 marks)

The material determines the properties the user actually experiences. Surface durability and hygiene: glass and melamine wipe clean and resist stains and moisture absolutely, oiled wood does not but can be repaired, and a scratched melamine face cannot be repaired at all. Thermal and chemical tolerance: a hot pan marks a lacquered wood top and is harmless on glass. Mass and handling: a solid oak top is a two-person lift and shifts the product from flat-pack to assembled logistics, which changes shipping cost and the buyer’s willingness to move house with it. Stiffness and permitted span: for a panel in bending the deflection goes as the cube of thickness, so a low-modulus material simply needs a thicker top or more frame. Perceived quality and lifetime: wood is refinished and inherited, particleboard is discarded when the edge swells, which sets the realistic service life at perhaps thirty years against seven. Safety: tempered glass fragments into blunt pieces rather than shards, which is why nothing else is acceptable for a glass top in a domestic kitchen. The material therefore fixes the market segment, the price point and the expected life before a single dimension is drawn.

C — How the material choice affects the manufacturing process (3 marks)

Choosing a material chooses a factory. Shaping: hardwood is sawn, planed, edge-glued and moulded on woodworking machinery with dust extraction; panel product is cut on a beam saw and edge-banded on a throughfeed line; glass is cut, ground, drilled and then tempered in a furnace, which is a different industry entirely and is almost always subcontracted. Joining: wood takes mortise-and-tenon or dowelled joints with adhesive cure time; panel takes knock-down cam fittings, which is why flat-pack furniture is made from it; glass takes no joint at all and must be clamped through an elastomer. Tolerance and process capability: wood moves after machining so tolerances must be generous and conditioning time allowed, whereas a tempered panel is fixed for ever at the moment it enters the furnace, so every hole must be right first time. Finishing: solid wood needs sanding, staining, sealing and curing — often the longest and most capital-intensive part of the line — while melamine-faced panel arrives finished and glass needs only washing. Yield and scrap: wood yields perhaps 60 % of the board bought, and the offcuts have little value; panel nests at above 90 %; glass scrap from a tempering failure is total. Volume sensitivity: all of this feeds the two-term cost model $c(n)=T/n+u$, and the material largely decides whether the route is tooling-heavy or labour-heavy.

D — A material-selection framework, applied to the computer case (6 marks)

The framework is Ashby’s four-step method, which is used here because it separates the constraints that must be satisfied from the objective that is to be optimised, and so cannot be argued into a preferred answer.

1. Translate. State the function, the constraints, the objective and the free variables. Function: a side panel of a desktop computer case, a rectangular panel of fixed in-plane dimensions loaded in bending. Constraints: it must not deflect visibly under a hand load, must not dent in shipping, must provide electromagnetic shielding, must not burn (UL 94 V-0 or non-combustible), and must conduct a little heat outward. Objective: minimise cost, with mass as a secondary objective. Free variable: the panel thickness.
2. Screen. Apply the constraints as hard filters. The electromagnetic-shielding requirement is the decisive one and it eliminates all unfilled polymers outright unless they are metallised, which adds a process step and a cost the objective is trying to avoid.
3. Rank. Derive the material index from the objective and the idealisation. For a panel of fixed in-plane dimensions, free thickness, ranked on stiffness at minimum mass the index is $M_1=E^{1/3}/\rho$; for dent resistance it is $M_2=\sigma_y^{1/2}/\rho$; and for stiffness at minimum cost, which is the stated objective, it is $M_3=E^{1/3}/(\rho\,C_m)$. Rank the survivors on $M_3$, using $M_1$ to read off the mass penalty.
4. Document. Take the two or three top-ranked candidates and check the things the indices cannot see: supply, minimum wall thickness for the process, finish, recyclability, and the process economics at the planned volume.

Given. Four candidate materials for the side panel, with the planned volume 250,000 cases per year. Sheet-metal stamping needs $85,000 of tooling at a variable cost of $6.40 per panel; injection moulding needs $145,000 at $4.10.

Find. The ranking of the four candidates on the correct panel indices, the mass and material cost of an equal-stiffness panel of each, and the recommended material and process.

1. Evaluate the stiffness index for the correct idealisation. A panel of free thickness ranks on $M_1=E^{1/3}/\rho$, giving 0.757 for steel, 1.538 for aluminium, 1.164 for PC/ABS and 1.965 for magnesium, so on mass-efficiency the order is $$\text{Mg}\;(1.965)\;>\;\text{Al}\;(1.538)\;>\;\text{PC/ABS}\;(1.164)\;>\;\text{steel}\;(0.757)$$Using the beam index $E^{1/2}/\rho$ by mistake gives 3.706, 3.122, 1.347 and 1.846 — which pushes steel above the polymer and reorders the shortlist. The exponent follows the idealisation, and a case panel is a panel, not a beam.
2. Convert the index into an equal-stiffness panel. At equal bending stiffness the required thickness scales as $t\propto E^{-1/3}$ and the mass as $m\propto\rho/E^{1/3}$. Taking a 1.00 mm steel panel of mass 1.00 as the reference, the aluminium panel is 1.44 mm and 0.492, the magnesium 1.67 mm and 0.385, and the PC/ABS 4.44 mm and 0.650. The polymer’s thickness is the first practical objection: a 4.4 mm moulded wall sinks, warps and takes about four times the cooling time of a 2 mm one.
3. Rank on the stated objective, which is cost. Multiplying each equal-stiffness mass by its material price gives 1.10, 2.07, 2.34 and 2.12 dollars for steel, aluminium, PC/ABS and magnesium, and equivalently$$M_3=\frac{E^{1/3}}{\rho\,C_m}:\quad 0.688\ \text{(steel)},\ 0.366\ \text{(Al)},\ 0.323\ \text{(PC/ABS)},\ 0.357\ \text{(Mg)}$$Steel wins on stiffness per dollar by a factor of nearly two, and it satisfies the shielding and fire constraints for free, which is precisely what the screening step predicted.
4. Settle the process at the planned volume. Comparing stamping with moulding through $c(n)=T/n+u$, the break-even is$$n^{*}=\frac{145{,}000-85{,}000}{6.40-4.10}=26{,}087\ \text{units}$$so at 250,000 per year moulding would be the cheaper process ($4.68 against $6.74 per panel) — but the polymer was already eliminated at the screening step, and adding conductive paint to recover the shielding costs more than the difference. Process economics decide between routes that survived screening; they do not resurrect a candidate that failed a constraint.
5. Document the recommendation. The panel is made from 0.8–1.0 mm cold-rolled steel, stamped and folded, with a PC/ABS front bezel where the requirement is appearance rather than shielding.$$\boxed{\text{Steel panel: }M_3=0.688\ \text{vs}\ 0.366\ \text{(Al)};\ \text{stamped, }6.74\ \text{per panel at }250{,}000/\text{yr}}$$in dollars. Aluminium is the correct answer only for a premium or portable chassis, where the 0.51 kg saved per panel is worth roughly a dollar of material — a trade the mass-market objective does not support but a workstation-class product does.