Page 90 - Shigley's Mechanical Engineering Design

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Materials 65
If the function is separable, which it often is, we can write Eq. (2–23) as

P = f 1 (F) · f 2 (G) · f 3 (M) (2–24)
For optimum design, we desire to maximize or minimize P. With regards to material
properties alone, this is done by maximizing or minimizing f 3 (M), called the material
efficiency coefficient.
For illustration, say we want to design a light, stiff, end-loaded cantilever beam with
a circular cross section. For this we will use the mass m of the beam for the performance
metric to minimize. The stiffness of the beam is related to its material and geometry. The
stiffness of a beam is given by k = F/δ, where F and δ are the end load and deflection,
respectively (see Chap. 4). The end deflection of an end-loaded cantilever beam is given
3
in Table A–9, beam 1, as δ = y max = (Fl )/(3EI), where E is Young’s modulus, I the
second moment of the area, and l the length of the beam. Thus, the stiffness is given by
F 3EI
k = = (2–25)
δ l 3
From Table A–18, the second moment of the area of a circular cross section is
π D 4 A 2
I = = (2–26)
64 4π
where D and A are the diameter and area of the cross section, respectively. Substituting
Eq. (2–26) in (2–25) and solving for A, we obtain

3 1/2
4πkl
A = (2–27)
3E
The mass of the beam is given by
m = Alρ (2–28)
Substituting Eq. (2–27) into (2–28) and rearranging yields

π 1/2 5/2 ρ
m = 2 (k )(l ) (2–29)
3 E 1/2
√
Equation (2–29) is of the form of Eq. (2–24). The term 2 π/3 is simply a constant and
√ 1/2
can be associated with any function, say f 1 (F). Thus, f 1 (F) = 2 π/3(k ) is the func-
tional requirement, stiffness; f 2 (G) = (l 5/2 ), the geometric parameter, length; and the
material efficiency coefficient
ρ
f 3 (M) = (2–30)
E 1/2
is the material property in terms of density and Young’s modulus. To minimize m we
want to minimize f 3 (M), or maximize
E 1/2
M = (2–31)
ρ
1
where M is called the material index, and β = . Returning to Fig. 2–16, draw lines of
2
various values of E 1/2 /ρ as shown in Fig. 2–17. Lines of increasing M move up and to
the left as shown. Thus, we see that good candidates for a light, stiff, end-loaded can-
tilever beam with a circular cross section are certain woods, composites, and ceramics.
Other limits/constraints may warrant further investigation. Say, for further illustra-
tion, the design requirements indicate that we need a Young’s modulus greater than