Dissemination of IT for the Promotion of Materials Science (DoITPoMS)

# Strength - Density selection map

Strength σf is the maximum stress which a material can support before failure, where failure is usually taken to be the end of the purely elastic regime. For ductile materials σf is the stress for the onset of plastic (irreversible) flow. Brittle materials break before they yield, so σf is the stress for the onset of brittle fracture.

Merit indices used in conjunction with these maps are:

$${{{\sigma _f}} \over \rho }$$, 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.

$${{{\sigma _f}^{2/3}} \over \rho }$$, 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.

The merit index $${{{\sigma _f}^{1/2}} \over \rho }$$ 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:

$$I = {{w{h^3}} \over {12}}$$

and the mass of the beam is:

$$m = whL\rho$$

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

$$h = {m \over {wL\rho }}$$

For beam bending, $$M = \kappa EI$$ and $$\kappa =$$$${{{\sigma _f}} \over {Eh}}$$ (Derivation), therefore,

$$M = {{{\sigma _f}I} \over h}$$

Substituting in the value for I gives:

$$M = {{{\sigma _f}w{h^2}} \over {12}}$$

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

$$M = {{{m^2}} \over {12w{L^2}}}\left( {{{{\sigma _f}} \over {{\rho ^2}}}} \right)$$

Hence maximising $${{{\sigma _f}} \over {{\rho ^2}}}$$ or $${{{\sigma _f}^{1/2}} \over \rho }$$ by moving the merit index line towards the top and the left of the above materials-selection map gives the material that is strongest in bending a beam of a given mass (given the conditions of fixed beam width and variable height).