11.3: Strength - Density Selection Map

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Strength \(σ_f\) 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 σ_f is the stress for the onset of plastic (irreversible) flow. Brittle materials break before they yield, so \(σ_f\) is the stress for the onset of brittle fracture.

Merit indices used in conjunction with these maps are:

\[\frac{\sigma_f}{\rho}\]

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

\[\frac{\sigma_f^{2/3}}{\rho}\]

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 \(\frac{\sigma_f^{1/2}}{\rho}\) for maximizing 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:

\[I = \frac{wh^3}{12}\]

and the mass of the beam is:

\[m=whL \rho\]

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

\[h = \frac{m}{wL \rho}\]

For beam bending, \(M = \kappa E I\) and \(\kappa = \frac{\sigma_f}{Eh}\) (Derivation), therefore,

\[ M = \frac{\sigma_f I}{h} \]

Substituting in the value for I gives:

\[M = \frac{\sigma_f w h^2}{12} \]

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

\[ M = \frac{m^2}{12 w L^2} \left ( \frac{\sigma_f}{\rho^2} \right ) \]

Hence maximising \( \frac{\sigma_f}{\rho^2} \) or \( \frac{\sigma_f^{1/2}}{\rho} \) 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).