Question 7 of 7: Material selection for a coat hanger

Question 7: Material selection for a coat hanger (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.

Part A — Three candidate materials and the challenges of each

The function is modest and the economics are brutal: a hanger is a commodity made in the hundreds of thousands, so a difference of five cents a unit decides the material. The duty is to carry a garment of up to about 3 kg without visible sag or permanent set, to present a shoulder profile that does not crease or mark the garment, to hook over a 25 to 32 mm rail, and to survive being dropped, trodden on and stacked.

Material 1 — drawn low-carbon steel wire, typically 2.5 to 3.2 mm diameter, galvanised or epoxy-coated. Challenges: the small circular section means high local contact pressure, so a bare wire hanger marks and can crease a shoulder, and knitwear distorts over it; the wire must be coated or it corrodes and stains the garment, and the coating is a second material and a second process; the twisted hook joint is the natural failure point and is difficult to inspect; and the section is so slender that although the material is stiff, the hanger relies on the wire triangle acting as a closed frame, so any opening of the twist collapses the geometry.

Material 2 — injection-moulded polypropylene (polystyrene is the cheaper but more brittle alternative). Challenges: the modulus is two orders of magnitude below steel, so the section must be made far deeper to reach the same stiffness, which is exactly what the arithmetic in part D shows; polypropylene creeps under sustained load at room temperature, so a heavy coat left on a thin hanger for a season produces permanent sag that no short-term test would reveal; unfilled polystyrene is notch-sensitive and snaps rather than bends; and the part must be designed for the process — uniform wall, draft, no thick sections that sink — which constrains the form more than a designer expects.

Material 3 — laminated hardwood, birch plywood or moulded bamboo. Challenges: cost is several times the alternatives, both in material and in conversion; wood is anisotropic and the strength across the grain is a small fraction of that along it, so the lay-up direction is a design variable and a badly oriented ply splits at the hook boss; moisture movement changes dimensions and can loosen the metal hook insert, which is itself a second material that compromises recyclability; and natural variation in the raw material means the process must be tolerant of a spread of properties.

Part B — How the choice of material impacts the design

The dominant effect is on section, and it can be quantified without a calculator. Treating the sloping arm as a cantilever of reach $L = 200$ mm carrying half a 30 N garment load, the tip deflection is

For a rectangular arm 10 mm wide and 6 mm deep, $I = 1.8 \times 10^{-10}$ m⁴, and the deflections are 1.06 mm in steel, 22.2 mm in birch laminate and 148 mm in polypropylene. Against a 3 mm limit for "no visible sag", only the steel passes at that section. The design response is not to abandon the other materials but to change the geometry: since $I$ goes as $h^3$, doubling the arm depth to 12 mm multiplies the second moment by eight and brings the laminate to 2.78 mm, inside the limit. This is why a wooden or plastic hanger is visibly chunky and a wire one is not — the bulk is not styling, it is the section the modulus demands.

Three further design consequences follow. Shoulder form: a moulded or laminated hanger has a broad, contoured shoulder because it needs the area anyway, and that broad shoulder is better for the garment, so the low-modulus materials get a functional advantage out of their structural handicap. Joints and features: a moulded hanger integrates the hook, the trouser bar and the notches as features at no extra cost, while wire needs a bend and a twist for each and wood needs an inserted hook. Failure mode: steel bends and gives warning, polypropylene creeps slowly, polystyrene and cross-grain wood break suddenly — and for a consumer product a benign failure mode is a real design requirement, not a footnote.

Part C — How the choice of material impacts the manufacturing process

Each material comes with its process, and the process brings its own cost structure, tolerance capability and minimum economic volume.

The structural point is that material choice does not merely select a process, it selects a cost structure, and the cost structure decides which material wins at which volume. Wire forming is tooling-light and change-friendly; injection moulding is tooling-heavy and change-hostile but has the lowest variable cost of the three; the laminate route is neither cheap to tool nor cheap to run and can only be justified by a price premium.

Comparing the routes on unit cost, $c(n) = T/n + u$, the crossover between wire forming and injection moulding is

Below that volume wire forming is cheaper — at 100 000 a year the three routes cost 0.175, 0.345 and 0.710 dollars respectively — and above it moulding takes over, costing 0.111 against 0.144 for wire at a million a year. The laminate route never wins on cost at any volume, which is the correct and useful conclusion: it is a premium product decision, and the 0.535 dollars a unit it gives away at 100 000 units is the premium the retail price must recover.

Part D — A framework for material selection, applied

Given. The three candidates, with the properties below, and a hanger arm loaded in bending. Properties are quoted in the units the material charts use: modulus in GPa, strength in MPa, density in Mg/m³ and material cost in Canadian dollars per kg.

Find. A defensible selection framework, and the ranking it produces for the coat hanger under three different objectives.

Approach. Use the four-step method of translate, screen, rank and seek supporting information. The ranking step needs a material index, which is derived by writing the objective as an equation, eliminating the free geometric variable using the constraint, and collecting the material properties that remain.

1. Translate the design requirement. The function is a beam loaded in bending. The constraints are a stiffness sufficient that tip deflection stays under 3 mm at 30 N, no permanent set at that load, no staining or snagging of the garment, a benign failure mode, and a maximum arm depth of about 12 mm for storage density. The objective is stated three different ways below, because which one is chosen decides the answer. The free variable is the section dimension.
2. Screen on the constraints, and eliminate before ranking. The deflection check does the work here: at the reference 10 × 6 mm section, steel deflects 1.06 mm, the laminate 22.2 mm and polypropylene 148 mm against the 3 mm limit. Polypropylene cannot be brought inside the limit within the 12 mm depth constraint and is eliminated as a structural arm at this section; the laminate reaches 2.78 mm once the depth is doubled to 12 mm, so it survives. Screening is done on constraints, never on the objective, and this step is where most of the candidates are properly removed.
3. Rank on a material index — and derive it rather than quoting it. For minimum mass at a given bending stiffness, with the section depth free, mass $m = A L \rho$ and stiffness $S \propto EI \propto E A^{2}$; eliminating $A$ between the two gives $m \propto \rho / E^{1/2}$, so the quantity to maximise is $$M_1 = \frac{E^{1/2}}{\rho}.$$ Evaluating: steel 1.85, polypropylene 1.35, birch laminate 4.65. The laminate wins by a factor of two and a half, which is why wooden hangers feel light for their bulk.
4. Re-rank under the other two objectives, because the objective decides the winner. For minimum mass at a given bending strength the same elimination gives $M_2 = \sigma_y^{2/3}/\rho$: steel 6.92, polypropylene 11.37, laminate 22.54 — the laminate again. But for minimum cost at a given bending stiffness the index carries the material price, $M_3 = E^{1/2}/(\rho C_m)$: steel 1.678, polypropylene 0.644, laminate 1.368, and now steel wins. That inversion is the whole lesson of the framework: $$\boxed{\ \text{the index you rank on decides the material you get.}\ }$$
5. Seek supporting information, then decide. The indices rank; they do not choose. Supporting information brings in what no index captures: polypropylene creeps under sustained room-temperature load, which no elastic index shows and which disqualifies it for heavy coats independently of the deflection screen; bare steel wire marks garments, so it needs a coating and is unsuitable for premium knitwear; the laminate needs a metal hook insert, so the mono-material argument that favours the polymer does not favour it either; and part C showed the laminate is never cost-competitive at any volume. Selection: coated drawn steel wire for the commodity product, on $M_3$ and on the volume argument of part C; birch laminate for a premium garment-care product, on $M_1$ and on the broad shoulder it affords; polypropylene only where the arm can be made deep and the load is light, such as a children's or retail display hanger.