91Ó°ÊÓ

(a) Using the procedure as outlined in Section 22.2 ascertain which of the metal alloys listed in Appendix B have torsional strength performance indices greater than \(10.0\left(\text { for } \tau_{f} \text { and } \rho \text { in units of } \mathrm{MPa} \text { and } \mathrm{g} / \mathrm{cm}^{3}\right.\) respectively \(),\) and, in addition, shear strengths greater than \(350 \mathrm{MPa}\). (b) Also using the cost database (Appendix C), conduct a cost analysis in the same manner as Section \(22.2 .\) For those materials that satisfy the criteria noted in part a, and, on the basis of this cost analysis, which material would you select for a solid cylindrical shaft? Why?

In a manner similar to the treatment of Section \(22.2,\) perform a stiffness- to-mass performance analysis on a solid cylindrical shaft that is subjected to a torsional stress. Use the same engineering materials that are listed in Table 22.1 . In addition, conduct a material cost analysis. Rank these materials both on the basis of mass of material required and material cost. For glass and carbon fiber-reinforced composites, assume that the shear module are 8.6 and \(9.2 \mathrm{GPa},\) respectively.

(a) Using the expression developed for stiffness performance index in Problem 22.D3(a) and data contained in Appen\(\operatorname{dix} \mathrm{B},\) determine stiffness performance indices for the following polymeric materials: high- density polyethylene, polypropylene, poly(vinyl chloride), polystyrene, polycarbonate, poly(methyl methacrylate),poly(ethylene terephthalate), polytetrafluoroethylene, and nylon \(6,6 .\) How do these values compare with those of the metallic materials? (Note: In Appendix B, where ranges of values are given, use average values.)(b) Now, using the cost database (Appen\(\operatorname{dix} C),\) conduct a cost analysis in the same manner as Section \(22.2 .\) Use cost data for the raw forms of these polymers. (c) Using the expression developed for strength performance index in Problem \(22 . \mathrm{D} 3(\mathrm{a})\) and data contained in Appendix B, determine strength performance indices for these same polymeric materials.

Consider the plate shown below that is supported at its ends and subjected to a force \(F\) that is uniformly distributed over the upper face as indicated. The deflection \(\delta\) at the \(L / 2\) position is given by the expression $$\delta=\frac{5 F L^{3}}{32 E w t^{3}}$$ Furthermore, the tensile stress at the underside and also at the \(L / 2\) location is equal to $$\sigma=\frac{3 F L}{4 w t^{2}}$$ (a) Derive stiffness and strength performance index expressions analogous to Equations 22.9 and 22.11 for this plate (Hint. solve for \(t\) in these two equations, and then substitute the resulting expressions into the mass equation, as expressed in terms of density and plate dimensions.) (b) From the properties database in Appendix \(\mathrm{B},\) select those metal alloys with stiffness performance indices greater than 1.40 (for \(E\) and \(\rho\) in units of \(\mathrm{GPa}\) and \(\mathrm{g} / \mathrm{cm}^{3}\) respectively). (c) Also using the cost database (Appendix C), conduct a cost analysis in the same manner as Section \(22.2 .\) Relative to this analysis and that in part b, which alloy would you select on a stiffness- per-mass basis? (d) Now select those metal alloys having strength performance indices greater than 5.0 (for \(\sigma_{y}\) and \(\rho\) in units of \(\mathrm{MPa}\) and \(\mathrm{g} / \mathrm{cm}^{3}\) respectively \(),\) and rank them from highest to lowest \(P\).(e) And, using the cost database, rank the materials in part d from least to most costly. Relative to this analysis and that in part d, which alloy would you select on a strength-per-mass basis? (f) Which material would you select if both stiffness and strength are to be considered relative to this application? Justify your choice.