Question 7 of 7: Material selection for a chair (15 marks)

Question 7 — Material selection for a chair (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.

Given. A chair seat rail of span $L = 420$ mm carrying the BIFMA X5.1 static seat load $F = 1136$ N at midspan, with a deflection limit of $L/200 = 2.1$ mm, idealised as a simply supported beam of free square section. Candidate material properties, and the two competing manufacturing routes, are tabulated below.

Process economics: fabricated steel tube (bend and weld), tooling $T_1 =$ CAD 6 000 and variable cost $u_1 =$ CAD 27.00 per chair; injection moulding, tooling $T_2 =$ CAD 85 000 and variable cost $u_2 =$ CAD 9.50 per chair.

Find. Three candidate materials with their challenges, the influence of the chair's final use and of the material choice on the manufacturing route, and a defensible selection framework applied to give a ranked recommendation — including the material index, the resulting rail mass for each candidate, and the process break-even volume.

Approach. Use Ashby's four-step method — translate, screen, rank, document — taking care to state the mechanical idealisation before choosing the index, because the exponent in the index follows the idealisation and choosing it wrongly reorders the shortlist.

A. Three materials and their challenges

1. Laminated hardwood (beech or maple). The traditional chair material, and on the mechanics it is still the best. Its challenges are variability and environment: mechanical properties are anisotropic and vary between boards and with grain direction, so design values must be characteristic (fifth-percentile) values rather than means; it is hygroscopic, so it moves with humidity and joints loosen through seasonal cycling; it is subject to fungal and insect attack when damp; it burns, which matters in institutional occupancies; and it cannot be repaired by welding, so a broken member is a scrapped member. Steam bending and lamination require skilled labour and long cycle times, and finishing is a multi-step wet process with its own emissions controls.

2. Steel tube (AISI 1020 or similar). Cheap, ductile, weldable, entirely predictable and available in a wide range of sections. Its challenges are mass and corrosion: it is by far the heaviest of the candidates at equal stiffness, as computed below, and it must be coated — powder coat, plate or paint — with the coating becoming a durability liability of its own, since a chipped coating on a chair used outdoors or wet-mopped becomes a rust site within a season. Welding introduces distortion and a heat-affected zone, weld spatter must be dressed before finishing, and tube-end joints need either coping or cast connectors, both of which add cost. Thermally it is cold to the touch, which is a genuine comfort issue for seating.

3. Glass-filled polypropylene (30 % by weight). The volume material of modern stacking seating, and the only candidate that can produce a seat, back and structure as a single moulding. Its challenges are stiffness, time-dependence and tooling. The modulus is thirty times below steel's, so sections must be deep and ribbed; it creeps under sustained load, so a design stress of roughly a quarter of the short-term strength is required if a seat is not to sag over years of occupation; it is notch-sensitive at low temperature, which matters for a chair stored in an unheated Canadian space; it degrades under ultraviolet light unless stabilised; the glass fibre orientates with the flow, making properties anisotropic and weld lines weak; and shrinkage, sink and warp must be designed against with uniform wall thickness. Above all, the tooling is a large fixed cost that must be committed before the first part exists.

B. How the final use of the chair drives the material choice

“A chair” is not a specification, and the intended use changes the answer completely. The mechanism is that use determines the constraints, and constraints eliminate candidates before any index is computed.

Domestic dining chair — low duty, appearance dominant, low volume per model: hardwood wins on appearance, repairability and mass. Office task chair — adjustability, eight-hour comfort, BIFMA X5.1 compliance and casters: a hybrid, with a steel or aluminium base and a moulded seat-and-back unit, because the function is mechanism-led. Institutional and school seating — abuse resistance, cleanability, fire performance, stacking, and very high volume: glass-filled polypropylene on a steel frame, and the fire requirement alone may eliminate untreated wood. Outdoor and patio seating — ultraviolet exposure, freeze–thaw, standing water and de-icing salt: uncoated hardwood and coated steel both become maintenance liabilities, and UV-stabilised polymer or anodised aluminium wins on a constraint rather than on an index. Healthcare seating — disinfectant compatibility and no fabric-covered crevices: seamless mouldings, with hardwood eliminated by the cleaning protocol rather than by mechanics. Heavy-duty or bariatric seating — a 300 kg rated load rather than the standard 1136 N: steel returns, because ductile failure and weldable repair matter more than mass.

Two further use-driven quantities cut across all of these. Volume decides whether tooling can be amortised at all, which is settled in part C. Stackability forces a moulded or formed monocoque and effectively eliminates a framed hardwood construction, and a chair that must stack forty high has a mass constraint imposed by the person who lifts the stack, not by the person who sits on it.

C. How the material choice drives the manufacturing process

Material and process are not chosen sequentially; each constrains the other, and the pairing is what actually gets selected. Wood dictates a subtractive and adhesive route — sawing, moulding, CNC profiling, steam bending or laminating in a former, then sanding and multi-coat finishing — with long cycle times, modest tooling, and a labour content that scales linearly with volume, which suits low and medium volumes and craft differentiation. Steel dictates a formative and joining route — tube bending on a mandrel bender, end coping, jig welding (MIG or resistance), dressing, degreasing and powder coating — with low tooling cost per model (a bend die and a weld jig), moderate labour and easy design change, so a new model can be introduced for a few thousand dollars. Thermoplastic dictates injection moulding: very high tooling cost, very low variable cost, cycle times of a minute or two, near-zero finishing because colour and texture are in the tool, but a design that is frozen the day the steel is cut, since a wall-thickness change means a new core.

The consequence is economic, and it is quantitative. With unit cost $c(n) = T/n + u$, the routes are equal at

$$n^{*} = \frac{T_2 - T_1}{u_1 - u_2} = \frac{85\,000 - 6\,000}{27.00 - 9.50} = \boxed{4\,514\ \text{chairs}}$$

Below that volume the fabricated route is cheaper — at 2 000 chairs, CAD 30.00 against CAD 52.00 — and above it the moulded route pulls away rapidly, reaching CAD 18.00 against CAD 27.60 at 10 000 chairs and CAD 12.90 against CAD 27.24 at 25 000. A single number therefore separates two entirely different businesses, and it must be computed before the material is chosen rather than after.

D. A selection framework, applied

The framework is Ashby's, in four steps, and the discipline of it lies in doing the steps in order.

1. Translate. State the function, the constraints, the objective and the free variables. Here: function — a seat rail carrying the occupant load in bending; constraints — must not deflect more than $L/200$ under 1136 N, must not yield, must meet BIFMA X5.1 and the fire and cleanability requirements of the chosen market, length fixed at 420 mm; objective — minimise mass (which is also minimising cost of material and of shipping); free variables — the material and the cross-section dimension.
   The idealisation must be stated here, because the index depends on it. A seat rail is a beam of free square section, not a panel of free thickness. For a beam, $I = a^{4}/12$ with $m = \rho a^{2}L$, so eliminating $a$ gives $m \propto \rho/E^{1/2}$ and the index to maximise is $M_1 = E^{1/2}/\rho$. Had the same part been idealised as a panel of fixed width and free thickness, the index would be $E^{1/3}/\rho$, which is a different ranking — on the panel index glass-filled polypropylene (0.0161) rises above aluminium (0.0152), whereas on the correct beam index aluminium (0.0973) is well ahead of the polymer (0.0686). Using the wrong exponent reorders the shortlist.
2. Screen. Apply the constraints as hard filters and delete, do not penalise. If the chair is institutional seating, an untreated hardwood fails the flame-spread requirement and leaves the list. If it is outdoor seating, unstabilised polypropylene and uncoated steel both leave the list. If it must be dishwasher-cleaned in a healthcare setting, hardwood leaves the list regardless of how well it ranks. This step is where most of the real decision is made, and no index can substitute for it.
3. Rank the survivors on the index, and then confirm with the actual mass. The required bending stiffness follows from the deflection constraint on a centrally loaded simply supported beam: $$\delta = \frac{FL^{3}}{48EI} \le \frac{L}{200} \;\Rightarrow\; (EI)_{\text{req}} = \frac{FL^{3}}{48\delta} = \frac{1136(0.420)^{3}}{48(0.0021)} = 835.0\ \text{N}\cdot\text{m}^{2}$$ For each candidate, $I = (EI)_{\text{req}}/E$, the square side is $a = (12I)^{1/4}$, and the rail mass is $m = \rho a^{2}L$. The results are collected in Table 7.1, and they follow the index exactly.
4. Document and decide. Check the ranking against everything the index cannot see — manufacturability of the resulting section, stress utilisation, cost, appearance, supply and end of life — and record the decision with its assumptions. Three checks matter here. First, the stress utilisations are all comfortable: the steel rail runs at 218 MPa against 350 (62 %), the beech at 29 MPa against 75 (39 %), the aluminium at 96 MPa against 275 (35 %) and the polypropylene at 15 MPa against 80 (19 %), so every candidate is stiffness-limited rather than strength-limited, which confirms $M_1$ as the right index. Second, the sections must be manufacturable: a 36 mm solid polypropylene rail is not a moulded part at all, and in practice the polymer would be a ribbed hollow section, which is exactly the shape-factor argument the simple index omits — the polymer's real position is better than the table shows, and the steel's is likewise better as a tube. Third, the creep check must be applied to the polymer, since 15 MPa sustained for years is close to the quarter-of-strength rule of thumb.

Recommendation. For a domestic or hospitality dining chair at a few thousand units per model, the framework selects laminated beech: it survives screening, it ranks first on the index by a wide margin at 0.249 kg per rail against 0.729 kg for steel, it is below the 4 514-unit break-even so tooling amortisation favours the low-tooling route, and its weaknesses — moisture movement and joint loosening — are addressed by lamination and mechanical joints rather than by a change of material. For institutional stacking seating at 25 000 units per model, screening removes untreated wood on fire performance and the volume is five times the break-even, so the selection becomes a glass-filled polypropylene seat-and-back moulding on a steel tube frame, at CAD 12.90 per chair against CAD 27.24 for the fabricated route. The framework produces two different answers because the constraints differ, and that is the correct behaviour: a selection method that returns the same material regardless of use is not being applied.