Consider a cantilever beam that is loaded with a tr verse force at its tip. The following materials are being considered: tungsten carbide, high-carbon heat-treated steel, polycarbonate polymer, aluminum alloy. Using the Ashby charts, recommend one or two of the materials for a design situation in which failure is by exceeding the strength of the material and it is desired to minimize the weight.
Question 2.28: Consider a cantilever beam that is loaded with a tr verse fo...
For strength,
\sigma=F l / Z=S (1)
where F l is the bending moment and Z is the section modulus [see Eq. (3-26b), p. 90]. The section modulus is strictly a function of the dimensions of the cross section and has the units in ^{3} (ips) or \mathrm{m}^{3} (SI). Thus, for a given cross section, Z=C(A)^{3 / 2}, where C is a number. For example, for a circular cross section, C=(4 \sqrt{\pi})^{-1}. Then, for strength, Eq. (1) is
\frac{F l}{C A^{3 / 2}}=S \quad \Rightarrow \quad A=\left(\frac{F l}{C S}\right)^{2 / 3} (2)
For mass,
Thus, f_{3}(M)=\rho / S^{2 / 3}, and maximize S^{2 / 3} / \rho \quad(\beta=2 / 3)
In Fig. (2-19), draw lines parallel to S^{2 / 3} / \rho
From the list of materials given, a higher strength aluminum alloy has the greatest potential, followed closely by high carbon heat-treated steel. Tungsten carbide is clearly not a good choice compared to the other candidate materials. .
Eq. (3-26b)
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