Materials and Structure

As an example of how the minimum weight criteria were obtained, consider a common structural shape, a cantilever beam supported at one end and loaded by some force, F, at the other end. In general, there will be some design parameters that must be met, for example the length, L, section, t x t, of the beam, and the load, F, it must stand without deflecting more that d. These quantities are related to the Young's modulus of the material by: d = 4FL³/Et⁴. The mass, M, of the beam depends on its density, r, and volume, Lt² : M = rLt².

Evaluating t² from the mass equation and putting this value in the deflection equation gives: M² = (4FL²/d)(r²/E). The material's properties are only in the "red" term, and the "blue" term contains only design constraints. To get the lowest mass system that will meet the restraints requires selecting one with the lowest value of (r²/E) or the maximum value of (E/r²).

From: Ashby and Jones,
"Engineering Materials 2,"
Pergamon (1986)

Similar procedures are used to select a material on the basis of resistance to plastic deformation. For this same shape the yield stress, s_y, is related to the applied force, F, by: F = s_yt³/6L. Using the expression for the mass to get rid of t in this equation gives: M = (6FL)^2/3(r/s_y^2/3)., where the material properties are again in the red term. This shows that to obtain the lowest mass system that will remain in the elastic range during use we need to select on the basis of the lowest value of (r/s_y^2/3), or the maximum value of (s_y/r^3/2). This expression also shows the importance of the material density.