Merit indices used in conjunction with these maps are:

, used to find the material giving the strongest strut or tie in tension, for a given mass with the cross-sectional area of the beam as the free parameter.

, used to find the material giving the strongest beam (i.e. that supporting the largest bending moment before the onset of plastic yielding or other failure on the surface of the beam) of a given mass, on bending a beam with a specified cross-sectional shape but free cross-sectional area.

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The merit index for maximising the strength of a beam under a bending load for a given mass, with a specified beam width, and unspecified height will now be derived:

Consider a beam of length L, width w and height h, subject to an end load F. The second moment of area is:

and the mass of the beam is:

which can be rearranged in terms of the free parameter h as:

For beam bending, and (Derivation), therefore,

Substituting in the value for I gives:

To find the desired merit index the free parameter h can then be eliminated from the equation giving:

Hence maximising or by moving the merit index line towards the top and the left of the above materials-selection map gives the material that is strongest for a given mass.

Silkworm and cocoon.

Looking at the materials-selection map, it can be seen that silk and cellulose are good materials for these applications. Silk is used in nature by silk worms to form their cocoons, and so must be strong and not easily breakable, as this would kill the silkworm, preventing it maturing into a moth and reproducing. Silkworms originate from China, India and Japan, and have been used to make silk by humans since at least 3,000 BC. Although silkworms only live for two months, they manage in this time to eat roughly 30,000 times their initial weight. It is estimated that 2,500 to 3,000 cocoons are needed to make just one yard of silk fabric, so despite silk being an excellent material for making fibres it is also expensive to produce. Cellulose is found in wood as the main fibre in the composite material, and hence must be strong so that when trees are bent the fibres will not break causing the tree trunk or branch to snap.

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