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Materials selection and design

A beam of square cross section with fixed length and bending stiffness is to be made as light as possible by freely varying the size of the cross section. Which performance index should be maximized, with E the Young's modulus and ρ the density?

  1. AE/ρ
  2. B√E/ρ
  3. C∛E/ρ
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Answer: B. √E/ρ

The deflection of a beam is inversely proportional to the flexural rigidity EI, and the second moment of area of a square section of side b is I = b⁴/12. To meet the specified stiffness, Eb⁴ must be constant, so b² ∝ 1/√E. The mass of the beam is m = ρb²L ∝ ρ/√E, so minimizing mass means choosing the material with the largest √E/ρ. E/ρ is the index for a tie limited by stiffness, used when the cross-sectional area enters directly. ∛E/ρ is for the bending stiffness of a plate (panel) of fixed width whose thickness alone is varied, coming from the stiffness being proportional to the cube of the thickness. In bending, stiffness increases rapidly as the section grows, so low-density materials are favored, which is why wood and CFRP are strong performers as beams.

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More Materials selection and design questions

  1. 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?
  2. 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?
  3. A tie of fixed length and tensile stiffness is made of steel (E = 210 GPa, ρ = 7.8 g/cm³, price per kg taken as 1) and of quasi-isotropic CFRP laminate (E = 70 GPa, ρ = 1.6 g/cm³, price per kg 20 times that of steel). If the cross-sectional area can be varied freely, about how many times the material cost of the steel tie is that of the CFRP tie?
  4. For a component used at a stress equal to its yield stress σy, the largest crack size that still lets it yield before brittle fracture can be estimated as a = (1/π)(K_IC/σy)² (geometry factor Y = 1). For a steel with σy = 800 MPa and K_IC = 120 MPa·√m, which value is closest to this critical crack length?
  5. For a typical gasoline passenger car, when the energy consumed over the life cycle from material production to disposal is compared stage by stage, which stage is usually the largest, and what materials selection policy follows from it?