Past exam of the materials science course of the University of Cambridge / 2022 / Paper 1 / 6 / c

Both outer faces of the cantilever are loaded, so use the pine's lower longitudinal compressive strength, $40\mathrm{MPa}$, rather than its $85\mathrm{MPa}$ tensile strength. Consistent SI units give $M_A$ values $(2.564,2.143,0.7273)\times 10^5{\mathrm{m}}^2{\mathrm{s}}^{-2}$ for steel, aluminium and pine respectively. Their $M_B$ values are $5.734,8.748,11.499\mathrm{P}{\mathrm{a}}^{1/2}{\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}$. Hence Design A: steel best, pine worst; Design B: pine best, steel worst. Aluminium is intermediate in both cases. These rankings concern the stated strength constraint; a toughness or deflection constraint could change feasibility.

At identical applied load and fixed geometric constraints, minimum mass is inversely proportional to the appropriate material performance index. ThereforeThe same percentages apply to weight under the same gravitational acceleration. It would be incorrect to compare densities alone, since the required beam dimensions also change.

On a logarithmic strength–density materials selection chart, constant $M_A$ gives ${\sigma }_f=M_A\rho$, a guideline of slope one; constant $M_B$ gives ${\sigma }_f=M_B^2{\rho }^2$, of slope two. Moving each guideline upwards selects better candidates. A suitably qualified longitudinal carbon-fibre-reinforced polymer is a promising candidate for both designs. Composite families occupy favourable high-strength/low-density regions in published course selection charts, but a beam must meet its compressive as well as tensile allowable.^[ref]

For scale, an explicitly illustrative candidate with ${\sigma }_{\lim }=1\mathrm{GPa}$ in both required loading senses and $\rho =1600\mathrm{kg}{\mathrm{m}}^{-3}$ would have $M_A=6.25\times 10^5{\mathrm{m}}^2{\mathrm{s}}^{-2}$ and $M_B=19.76\mathrm{P}{\mathrm{a}}^{1/2}{\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}$, exceeding the best supplied materials. These are assumed screening values, not measured properties of a specified product.