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Materials selection and design

Beams of square cross section designed for the same length and bending stiffness are made of steel (E = 210 GPa, ρ = 7.8 g/cm³) and an aluminum alloy (E = 70 GPa, ρ = 2.7 g/cm³). If the size of the cross section can be varied freely, about how many times the mass of the steel beam is the mass of the aluminum alloy beam?

  1. AAbout 0.60 times
  2. BAbout 0.35 times
  3. CAbout 1.04 times
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Answer: A. About 0.60 times

The mass of a beam with specified stiffness is proportional to ρ/√E. For the aluminum alloy it is 2.7/√70 = 2.7/8.37 ≈ 0.323, and for steel 7.8/√210 = 7.8/14.5 ≈ 0.538, so the ratio is 0.323/0.538 ≈ 0.60, about 0.60 times. The aluminum alloy beam needs a thicker section because its Young's modulus is one third, but thanks to its low density it is about 40% lighter overall. About 0.35 times looks only at the density ratio (2.7/7.8), ignoring that the section must be thicker to match stiffness. About 1.04 times is the value when mass is proportional to ρ/E, as for a tie in tension; for a tie, aluminum alloy and steel weigh almost the same.

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