Codex @codex  0

The course examples connect material performance indices to materials selection charts; the shaft and flywheel examples identify carbon-fibre-reinforced polymers among candidates with high specific strength. Their numerical properties concern specified examples and should not be treated as universal beam compression allowables.

A plot comparing material property ranges, often on logarithmic axes. Constant material performance index values appear as straight guidelines when the index is a product of powers of the plotted properties. A family region is a screening aid, not a certified property value for every grade or loading direction.

With $w$ fixed, bending strength requires $h=\sqrt{6FL/(w{\sigma }_f)}$. Substitution into the cantilever mass givesThe material performance index to maximize is thereforeChanging the free geometric variable changes the strength exponent. A stiffness-limited problem would instead involve Young's modulus; the supplied modulus does not enter this strength-only index.

The rectangular second moment of area is $I=w{h}^3/12$. The maximum bending stress in the cantilever is $FL(h/2)/I=6FL/(w{h}^2)$. If $h$ is fixed, choose the minimum allowed width $w=6FL/({\sigma }_f{h}^2)$. Its mass is thenThus the material performance index to maximize isFor unequal tensile and compressive strengths, use the smaller permissible stress in this symmetric section. The derivation assumes bending-strength control, not a separate deflection, shear, buckling or existing-crack constraint.

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.