Question 7 of 7: Material Selection for a Bicycle Crank Arm

Question 7: Material Selection for a Bicycle Crank Arm (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 their effect on the bicycle

6061-T6 aluminium alloy, cold forged. The industry default for everything from mid-range to professional cranks. Forging gives a favourable grain flow around the spindle boss, where the stress concentration is worst, and the alloy is readily anodised for corrosion protection. On the bicycle it gives a light, stiff crank at moderate cost; its weakness is that aluminium has no true fatigue limit, so every stress cycle consumes life and the crank has a finite, if long, service life. Riders experience it as a well-balanced crank that eventually needs replacing rather than one that lasts for ever.

4130 chromium-molybdenum steel, welded or cold formed. The traditional choice, still used on utility, touring and cargo bicycles. It is by far the heaviest of the three at equal stiffness, which the rider feels as rotating mass at the crank radius — more effort to accelerate, though negligible at steady speed. In exchange it is the most forgiving material in service: it has a genuine fatigue limit below which life is effectively infinite, it deforms visibly before it fails rather than fracturing without warning, it can be welded and straightened in any workshop in the world, and it is the cheapest of the three.

Carbon-fibre-reinforced polymer (CFRP), moulded with a bonded aluminium spindle and pedal insert. The performance choice. It is dramatically the lightest at equal stiffness and the laminate can be tailored so that the crank is stiff in the pedalling plane and more compliant laterally, which is a design freedom the metals do not offer. The rider experiences a noticeably more responsive drivetrain. Against that, it is the most expensive by a wide margin, it is vulnerable to impact and to point loading from a chain drop or a rock strike, the damage is often invisible from the surface, and it cannot be inspected or repaired in the field — a crank that has been hit must be replaced rather than assessed.

Part B — Five design criteria for a crank arm

1. Fatigue strength under fully reversed bending. The governing load case. The crank sees a bending moment that reverses every revolution, of the order of a million cycles per season, and it is a safety-critical part: failure at the spindle boss puts the rider on the ground. The Canadian and international requirement is the fatigue test of EN ISO 4210-8, and this criterion, not static strength, sets the section.
2. Bending and torsional stiffness. Pedal-end deflection under load is felt directly as lost power and as a vague pedalling feel. A tip deflection limit of about 2 mm at peak load is a reasonable target for a performance crank.
3. Mass. The crank is rotating mass at a large radius and is reciprocating, so it counts more than an equal mass on the frame — this is the criterion that separates the markets in Part D.
4. Cost at the intended production volume, including the tooling amortisation, which for a forged crank is substantial and for a moulded composite crank is larger still.
5. Damage tolerance, corrosion resistance and inspectability in service. The part lives outdoors, is exposed to road salt on Canadian winter roads, and is struck by rocks, chains and pedals. Whether a damaged crank can be identified before it fails is a design property of the material choice.

A sixth criterion, the interface geometry at the square-taper spindle and the pedal thread, is fixed by compatibility standards rather than chosen, and it constrains the minimum section at both ends.

Part C — A material-selection framework, and its application

The framework is Ashby’s four-step procedure — translate, screen, rank, document — with one addition that the marks usually turn on: after ranking on the stiffness index, the stress utilisation must be computed to check that stiffness really is the binding constraint. Assuming it is, is the standard error.

Step 1, translate. The function is a cantilever beam carrying the pedal load to the spindle. The objective is to minimise mass. The constraints are a specified tip stiffness, no fatigue failure at the required life, and a length fixed at 175 mm by ergonomics. The free variable is the cross-section, which is why this is a beam of free section and not a panel — the distinction fixes the exponents in the index and is the commonest place to go wrong.

Step 2, screen. Eliminate on constraints that no amount of index performance can rescue: the material must survive outdoor exposure and road salt (which removes plain unprotected carbon steel unless it is plated); it must be joinable to a steel pedal thread and a steel spindle interface without galvanic attack (which is why the CFRP option carries bonded metallic inserts and cannot be all-composite); and it must be available in a form the chosen process can use at volume.

Step 3, rank on the derived index. For a beam of free section, minimum mass at specified bending stiffness ranks materials on $M_1=E^{1/2}/\rho$, and at specified bending strength on $M_2=\sigma_y^{2/3}/\rho$. Both are derived below by sizing the actual part rather than quoted, because the sizing is what produces the utilisation check.

Given.

Find. The section, mass and stress utilisation of each candidate at equal stiffness, hence which index governs and which material wins.

1. Fix the stiffness requirement as a required flexural rigidity. For a cantilever with an end load, $\delta=FL^{3}/(3EI)$, so $$(EI)_{\text{req}}=\frac{FL^{3}}{3\delta}=\frac{1250\times 0.175^{3}}{3\times 0.002}=1116.5\ \text{N}\cdot\text{m}^{2}$$ and the same requirement applies to every candidate, which is what makes the comparison fair. The bending moment at the spindle is $M=FL=1250\times 0.175=218.75$ N·m for all three.
2. Size each section from that rigidity. With $h=3b$, the second moment of area is $I=bh^{3}/12=2.25\,b^{4}$ and the section modulus is $Z=I/(h/2)=1.5\,b^{3}$. Setting $I=(EI)_{\text{req}}/E$ and solving $b=(I/2.25)^{1/4}$ gives the widths in the table below; the mass follows from $m=\rho\,(3b^{2})\,L$.
3. Check the stress utilisation before believing the ranking. The bending stress at the spindle is $\sigma=M/Z$, and the utilisation is $\sigma/\sigma_y$. This is the step that decides which index applies, and it changes the answer for one of the three candidates.

The internal check on the arithmetic is worth stating because it confirms the index is being used correctly: the CFRP-to-aluminium mass ratio is $70.7/120.2=0.588$, and the inverse ratio of the stiffness indices is $3.077/5.229=0.588$. The index and the explicit sizing agree exactly, as they must, because the index is nothing more than the sizing calculation with the geometry eliminated.

What the utilisation check changes. Aluminium and CFRP are comfortably stiffness-limited at 0.68 and 0.32 of yield, so for those two the $E^{1/2}/\rho$ ranking stands. Steel is not: at 0.92 of yield it has no usable margin for a fully reversed fatigue load, so it is simultaneously strength-critical and must be re-sized on strength. Taking a design allowable of $\sigma_y/1.5=306.7$ MPa requires $Z=M/\sigma_{\text{allow}}=7.13\times10^{-7}$ m³, hence $b=7.81$ mm and a mass of 251 g rather than 203 g. Steel is therefore 24 per cent heavier than the stiffness ranking suggested, and the gap to the other two widens. Had the utilisation been the other way round — a candidate comfortable on stiffness but marginal on strength — the $\sigma_y^{2/3}/\rho$ ranking would have had to be shown as well, because it reorders the shortlist.

An honest caveat that a complete answer must include. Even the aluminium section, at 187 MPa, sits well above the fatigue strength of 6061-T6 at $10^{7}$ cycles, which is of the order of 110 MPa. A solid rectangle is therefore not a viable crank in any of these materials, and that is the correct conclusion rather than a failure of the analysis. Real cranks are forged or moulded with an I-section or a hollow section whose shape factor is roughly 2.5 to 3, which raises the section modulus at constant area and brings the stress below the fatigue limit. The material ranking is unaffected — the shape factor is available to all three — but the sizing must be repeated on the real section before any drawing is issued.

Step 4, document, and the selection. On the framework as stated — minimum mass at specified stiffness, with the strength check applied — the ranking is CFRP first at 70.7 g, aluminium second at 120.2 g, and steel third at 251 g once it is sized properly. But the objective as stated in Part C is "the best material to make the component", and mass is only one of the five criteria from Part B. Ranking on cost, damage tolerance and inspectability inverts the order, which is exactly why Part D exists: there is no single best material, only a best material for a stated market.

Part D — Adjusting the framework for three markets

The framework is adjusted not by changing the physics but by changing the objective function: the constraints and the indices stay where they are, and the weights on the Part B criteria change to reflect what each market is actually buying. The weights are fixed before the ratings are applied, which is the discipline that keeps a weighted matrix honest, and each material is rated 1 to 5 on each criterion from the engineering evidence above.

1. Score each material in each market. The weighted score is $S=\sum_i w_i r_i$. For the commuter market the aluminium score is $$S=0.10(4)+0.10(4)+0.25(4)+0.35(4)+0.15(4)+0.05(4)=4.00$$ and repeating for the other combinations gives the results below. The winners are $\boxed{\text{steel for commuters, CFRP for road racing, aluminium for mountain biking}}$.

Reading the result. Three observations carry the marks. First, the physics never changed — the indices, the sizing and the ratings are identical in all three columns — and yet the selection changed completely; the material decision was never a materials question alone, it was a question about the objective. Second, the margins are narrow: aluminium loses the commuter market by 0.30 and the racing market by 0.20, and a 0.05 shift in a single weight would flip either. That is a sensitivity result, and it should be stated rather than hidden, because it means aluminium is the robust choice: it is never the best and never worse than second, which is precisely why forged 6061 dominates the real mid-market across all three segments. A manufacturer supplying all three with one crank would choose aluminium and accept losing the extremes. Third, the mountain result depends heavily on damage tolerance at a weight of 0.30, which is the criterion CFRP fails not because it is weak but because its damage is invisible — an inspectability argument rather than a strength one, and the kind of engineering judgement a material index cannot capture.