A flat panel whose thickness is chosen to give a fixed bending stiffness is changed from an aluminum alloy (Young's modulus 70 GPa, density 2.70 g/cm³) to a magnesium alloy (Young's modulus 45 GPa, density 1.74 g/cm³). If the area of the panel is the same and the bending stiffness is proportional to E × (thickness)³, about how many times the mass of the aluminum alloy panel is the mass of the magnesium alloy panel?

Answer: A. About 0.75 times

To keep the bending stiffness E·t³ the same, the thickness must be increased by (70/45)^(1/3) ≈ 1.16 times. Mass is proportional to thickness and density, so the ratio is 1.16 × 1.74/2.70 ≈ 0.75, about 25% lighter. About 0.64 times compares only the densities without changing the thickness, which leaves the stiffness too low. About 1.0 times assumes the stiffness is proportional to Young's modulus and increases the thickness by 70/45; that is how members governed by tensile stiffness compare. E/ρ is about 25 GPa/(g/cm³) for Mg, Al, Ti and steel alike, so they give no weight saving in tensile stiffness, but in panels whose thickness can be increased, the lower-density metals have the advantage. This is one reason magnesium alloy die castings are used for thin housings and automotive interior parts.