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Materials selection and design

On a log–log materials selection chart (Ashby chart) with Young's modulus E on the vertical axis and density ρ on the horizontal axis, what is the slope of the guideline along which the performance index for a light, stiff beam, √E/ρ, is constant?

  1. A1
  2. B2
  3. C1/2
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Answer: B. 2

Setting √E/ρ = C (constant) gives E = C²ρ², and taking logarithms of both sides gives log E = 2 log ρ + 2 log C, a straight line of slope 2. Materials on a line parallel to it perform equally, and the farther the line is shifted toward the upper left, the larger √E/ρ, that is, the lighter a beam of the same stiffness can be made. Slope 1 is the line of constant E/ρ (tie), and slope 3 is the line of constant ∛E/ρ (panel). Slope 1/2 is the mistake of swapping the axes and reading the slope of log ρ = (1/2) log E + constant. On a log–log chart, differences in the exponents of performance indices appear as differences in slope, so a single chart lets you read off the ranking of materials for each loading mode.

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