Chapter 22 Materials Selection and Design Considerations

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For a cylindrical shaft of length L and radius r that is stressed in torsion, expression for mass in terms of density and shear strength of the shaft material Strength performance index expression for a torsionally stressed cylindrical shaft 22.2 Strength Considerations—Torsionally Stressed Shaft • W89 It is now necessary to take into consideration material mass. The mass m of any given quantity of material is just the product of its density and volume. Since the volume of a cylinder is just pr then 2 (r) L, m � pr 2 Lr or, the radius of the shaft in terms of its mass is just m r � A pLr Substitution of this r expression into Equation 22.4 leads to tf N � 2Mt m p a A pLr b 3 pL � 2Mt B 3r3 m3 Solving this expression for the mass m yields m � 12NM t2 2�3 1p 1�3 L2 a r (22.5) (22.6) (22.7) (22.8) The parameters on the right-hand side of this equation are grouped into three sets of parentheses.Those contained within the first set (i.e., N and ) relate to the safe functioning of the shaft.Within the second parentheses is L, a geometric parameter. Finally, the material properties of density and strength are contained within the last set. The upshot of Equation 22.8 is that the best materials to be used for a light shaft that can safely sustain a specified twisting moment are those having low r�t ratios. In terms of material suitability, it is sometimes preferable to work with what is termed a performance index, P, which is just the reciprocal of this ratio; that is, 2�3 Mt f P � t2�3 f r t2�3 f (22.9) In this context we want to utilize a material having a large performance index. At this point it becomes necessary to examine the performance indices of a variety of potential materials. This procedure is expedited by the utilization of what are termed materials selection charts. 1 These are plots of the values of one material property versus those of another property. Both axes are scaled logarithmically and usually span about five orders of magnitude, so as to include the properties of virtually all materials. For example, for our problem, the chart of interest is logarithm of strength versus logarithm of density, which is shown in Figure 22.2. 2 It may be noted on this plot that materials of a particular type (e.g., woods, engineering polymers, etc.) cluster together and are enclosed within an envelope delineated with a bold line. Subclasses within these clusters are enclosed using finer lines. 1 A comprehensive collection of these charts may be found in M. F. Ashby, Materials Selection in Mechanical Design, 2nd edition, Butterworth-Heinemann, Woburn, UK, 2002. 2 Strength for metals and polymers is taken as yield strength, for ceramics and glasses, compressive strength, for elastomers, tear strength, and for composites, tensile failure strength. b
W90 • Chapter 22 / Materials Selection and Design Considerations Strength (MPa) 10,000 1000 100 10 1 P = 100 P = 30 P = 10 Cork P = 3 Balsa Balsa Parallel to Grain Polymer foams Engineering composites Ash Oak Pine Fir Ash Woods Oak Pine Fir Perpendicular to Grain Wood Products LDPE Soft Butyl PP PS Engineering ceramics Nylons PMMA Glasses MEL PVC Epoxies Polyesters HDPE PTFE PU Silicone CFRP GFRP UNIPLY KFRP CFRPBe GFRP Laminates KFRP Elastomers 0.1 0.1 0.3 1 3 10 30 B Si Mg Alloys Density (Mg /m 3 ) Engineering polymers Porous ceramics Figure 22.2 Strength-versus-density materials selection chart. Design guidelines for performance indices of 3, 10, 30, and 100 (MPa) 2�3 m 3 /Mg have been constructed, all having 3 a slope of 2. (Adapted from M. F. Ashby, Materials Selection in Mechanical Design. Copyright © 1992. Reprinted by permission of Butterworth-Heinemann Ltd.) Now, taking the logarithm of both sides of Equation 22.9 and rearranging yields log t f � 3 2 log r � 3 2 log P Diamond Si3N4 Sialons Al2O3 ZrO2 Cermets MgO Ge Engineering alloys (22.10) This expression tells us that a plot of log �f versus log will yield a family of straight and parallel lines all having a slope of each line in the family corresponds to a different performance index, P. These lines are termed design guidelines, and four have been included in Figure 22.2 for P values of 3, 10, 30, and 100 (MPa) All materials that lie on one of these lines will perform equally well in terms of 2�3 r 3 2; 3 m /Mg. SiC Pottery Ti Alloys Cement Concrete Al Alloys Stone, Rock Engineering alloys Steels Cast Irons Zn Alloys Lead Alloys Ni Alloys Cu Alloys W Alloys Mo Alloys
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Chapter 22 Materials Selection and Design Considerations

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