Materials Selection for a Torsionally Stressed Cylindrical Shaft

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1 Case Study 1 (CS1) Materials Selection or a Torsionally Stressed Cylindrical Shat Learning Objectives Ater studying this case study, you should be able to do the ollowing: 1. Briely describe how the strength perormance index or a solid cylindrical shat is determined. 2. Explain the manner in which materials selection charts are employed in the materials selection process. 3. Explain how actors such as a material s cost, stiness, and atigue behavior aect the materials selection process. CS1.1 CS1.2 INTRODUCTION Selection o the appropriate material is an important consideration in engineering design; that is, or some application, choosing a material having a desirable or optimum property or combination o properties. Selection o the proper material can reduce costs and improve perormance. Elements o this materials selection process involve deciding on the constraints o the problem and, rom these, establishing criteria that can be used in materials selection to maximize perormance. The component or structural element we have chosen to discuss is a solid cylindrical shat that is subjected to a torsional stress. Strength o the shat will be considered in detail, and criteria will be developed or maximizing strength with respect to both minimum material mass and minimum cost. Other parameters and properties that may be important in this selection process are also discussed briely. STRENGTH CONSIDERATIONS For this portion o the design problem, we will establish a criterion or selection o light and strong materials or the shat. We will assume that the twisting moment and length o the shat are speciied, whereas the radius (or cross-sectional area) may be varied. We develop an expression or the mass o material required in terms o twisting moment, shat length, and density and strength o the material. Using this expression, it will be possible to evaluate the perormance that is, maximize the strength o the torsionally stressed shat with respect to mass and, in addition, relative to material cost. Consider the cylindrical shat o length L and radius r, as shown in Figure CS1.1. The application o twisting moment (or torque) M t produces an angle o twist. Shear stress at radius r is deined by the equation t M t r J Here, J is the polar moment o inertia, which or a solid cylinder is (CS1.1) J pr4 2 (CS1.2) CS1.1

2 CS1.2 Case Study 1 / Materials Selection or a Torsionally Stressed Cylindrical Shat M t r Figure CS1.1 A solid cylindrical shat that experiences an angle o twist in response to the application o a twisting moment M t. L Thus, t 2M t (CS1.3) pr 3 A sae design calls or the shat to be able to sustain some twisting moment without racture. In order to establish a materials selection criterion or a light and strong material, we replace the shear stress in Equation CS1.3 with the shear strength o the material t, divided by a actor o saety N, as t (CS1.4) N 2M t pr 3 It is now necessary to take into consideration material mass. The mass m o any given quantity o material is just the product o its density ( r) and volume. Since the volume o a cylinder is just pr 2 L, then m pr 2 Lr (CS1.5) or, the radius o the shat in terms o its mass is just m r (CS1.6) A plr Substituting this r expression into Equation CS1.4 leads to t N 2M t 3 m p a A plr b For a cylindrical shat o length L and radius r that is stressed in torsion, expression or mass in terms o density and shear strength o the shat material Strength perormance index expression or a torsionally stressed cylindrical shat 2M t A pl 3 r 3 Solving this expression or the mass m yields m 12NM t p 1 3 L2a r (CS1.7) (CS1.8) The parameters on the right-hand side o this equation are grouped into three sets o parentheses.those contained within the irst set (i.e., N and M t ) relate to the sae unctioning o the shat. Within the second parentheses is L, a geometric parameter. Finally, the material properties o density and strength are contained within the last set. The upshot o Equation CS1.8 is that the best materials to be used or a light shat that can saely sustain a speciied twisting moment are those having low ratios. In terms o material suitability, it is sometimes preerable to work with what is termed a perormance index, P, which is just the reciprocal o this ratio; that is, P t2 3 r In this context, we want to use a material having a large perormance index. m 3 t 2 3 b r t 2 3 (CS1.9)

3 CS1.2 Strength Considerations CS1.3 At this point, it becomes necessary to examine the perormance indices o a variety o potential materials. This procedure is expedited by the use o materials selection charts. 1 These are plots o the values o one material property versus those o another property. Both axes are scaled logarithmically and usually span about ive orders o magnitude, so as to include the properties o virtually all materials. For example, or our problem, the chart o interest is logarithm o strength versus logarithm o density, which is shown in Figure CS It may be noted on this plot that materials o 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 iner lines. Now, taking the logarithm o both sides o Equation CS1.9 and rearranging yields log t 3 2 log r 3 2 log P (CS1.10) t This expression tells us that a plot o log versus log r will yield a amily o straight 3 and parallel lines all having a slope o 2; each line in the amily corresponds to a dierent perormance index, P. These lines are termed design guidelines, and our have been included in Figure CS1.2 or P values o 3, 10, 30, and 100 (MPa) 2 3 m 3 /Mg. All materials that lie on one o these lines will perorm equally well in terms o strengthper-mass basis; materials whose positions lie above a particular line will have higher perormance indices, whereas those lying below will exhibit poorer perormances. For example, a material on the P 30 line will yield the same strength with one-third the mass as another material that lies along the P 10 line. The selection process now involves choosing one o these lines, a selection line that includes some subset o these materials; or the sake o argument, let us pick P 10 (MPa) 2 3 m 3 /Mg, which is represented in Figure CS1.3. Materials lying along this line or above it are in the search region o the diagram and are possible candidates or this rotating shat. These include wood products, some plastics, a number o engineering alloys, the engineering composites, and glasses and engineering ceramics. On the basis o racture toughness considerations, the engineering ceramics and glasses are ruled out as possibilities. Let us now impose a urther constraint on the problem namely, that the strength o the shat must equal or exceed 300 MPa (43,500 psi). This may be represented on the materials selection chart by a horizontal line constructed at 300 MPa, Figure CS1.3. Now the search region is urther restricted to the area above both o these lines. Thus, all wood products, all engineering polymers, other engineering alloys (viz., Mg and some Al alloys), and some engineering composites are eliminated as candidates; steels, titanium alloys, high-strength aluminum alloys, and the engineering composites remain as possibilities. At this point, we are in a position to evaluate and compare the strength perormance behavior o speciic materials. Table CS1.1 presents the density, strength, and strength perormance index or three engineering alloys and two engineering composites, which were deemed acceptable candidates rom the analysis using the materials selection chart. In this table, strength was taken as 0.6 times the tensile 1 A comprehensive collection o these charts may be ound in M. F. Ashby, Materials Selection in Mechanical Design, 3rd edition, Butterworth-Heinemann, Oxord, Strength or metals and polymers is taken as yield strength; or ceramics and glasses, compressive strength; or elastomers, tear strength; and or composites, tensile ailure strength.

4 CS1.4 Case Study 1 / Materials Selection or a Torsionally Stressed Cylindrical Shat 10,000 ceramics Glasses B Si SiC Diamond Si 3 N 4 Sialons Al 2 O 3 ZrO2 MgO Ge Cermets alloys Strength (MPa) P = 100 P = 30 Balsa Balsa Parallel to Grain composites Ash Oak Pine Fir Ash Woods Oak Pine Fir Perpendicular to Grain Wood Products LDPE Sot Butyl PP PS Nylons PMMA MEL PVC Epoxies Polyesters HDPE PTFE PU Silicone CFRP GFRP UNIPLY KFRP CFRPBe GFRP Laminates KFRP Pottery Mg Cement Concrete polymers Ti Al Stone, Rock alloys Porous ceramics Steels Cast Irons Zn Lead Ni Cu W Mo 1 P = 10 Cork Polymer oams Elastomers P = Density (Mg /m 3 ) Figure CS1.2 Strength versus density materials selection chart. Design guidelines or perormance indices o 3, 10, 30, and 100 (MPa) 2 3 m 3 /Mg have been constructed, all 3 having a slope o 2. (Adapted rom M. F. Ashby, Materials Selection in Mechanical Design. Copyright Reprinted by permission o Butterworth-Heinemann Ltd.) yield strength (or the alloys) and 0.6 times the tensile strength (or the composites); these approximations were necessary because we are concerned with strength in torsion, and torsional strengths are not readily available. Furthermore, or the two engineering composites, it is assumed that the continuous and aligned glass and carbon ibers are wound in a helical ashion (Figure o Introduction;Figure o Fundamentals), and at a 45 angle reerenced to the shat axis. The ive materials in Table CS1.1 are ranked according to strength perormance index, rom highest to lowest: carbon iber reinorced and glass iber reinorced composites, ollowed by aluminum, titanium, and 4340 steel alloys.

5 CS1.2 Strength Considerations CS1.5 10,000 ceramics Glasses B Si SiC Diamond Si 3 N 4 Sialons Al 2 O 3 ZrO2 MgO Ge Cermets alloys Strength (MPa) MPa Parallel to Grain Ash Oak Pine Fir composites Wood Products Balsa Ash Woods Oak Pine Fir P = 10 Perpendicular to Grain (MPa) 2/3 m 3 /Mg LDPE Sot Balsa Butyl PP PS CFRP GFRP UNIPLY KFRP CFRPBe GFRP Laminates KFRP Nylons PMMA MEL PVC Epoxies Polyesters HDPE PTFE PU Silicone Pottery Mg Cement Concrete polymers Ti Al Stone, Rock alloys Porous ceramics Steels Cast Irons Zn Lead Ni Cu W Mo Elastomers 1 Cork Polymer oams Density (Mg /m 3 ) Figure CS1.3 Strength versus density materials selection chart. Materials within the shaded region are acceptable candidates or a solid cylindrical shat that has a mass-strength perormance index in excess o 10 (MPa) 2 3 m 3 /Mg and a strength o at least 300 MPa (43,500 psi). (Adapted rom M. F. Ashby, Materials Selection in Mechanical Design. Copyright Reprinted by permission o Butterworth-Heinemann Ltd.) Materials cost is another important consideration in the selection process. In real-lie engineering situations, the economics o the application is oten the overriding issue and will normally dictate the material o choice. One way to determine materials cost is by taking the product o the price (on a per-unit mass basis) and the required mass o material. Cost considerations or these ive remaining candidate materials steel, aluminum, and titanium alloys, and two engineering composites are presented in Table CS1.2. In the irst column is tabulated r t 2 3. The next column lists the approximate relative cost, denoted as c; this parameter is simply the per-unit mass cost o material

6 CS1.6 Case Study 1 / Materials Selection or a Torsionally Stressed Cylindrical Shat Table CS1.1 Density ( ), Strength ( ), and Strength Perormance Index (P) or Five Materials P Material (Mg m 3 ) (MPa) [(MPa) 2/3 m 3 Mg] Carbon iber reinorced composite (0.65 iber raction) a Glass iber reinorced composite (0.65 iber raction) a Aluminum alloy (2024-T6) Titanium alloy (Ti-6Al-4V) Steel (oil-quenched and tempered) a The ibers in these composites are continuous, aligned, and wound in a helical ashion at a 45 angle relative to the shat axis. T 2 3 divided by the per-unit mass cost or low-carbon steel, one o the common engineering materials. The underlying rationale or using c is that although the price o a speciic material will vary over time, the price ratio between that material and another will most likely change more slowly. Finally, the right-hand column o Table CS1.2 shows the product o c and r t 2 3. This product provides a comparison o the materials on the basis o the cost o materials or a cylindrical shat that would not racture in response to the twisting moment M t. We use this product inasmuch as r t 2 3 is proportional to the mass o material required (Equation CS1.8) and c is the relative cost on a per-unit mass basis. Now the most economical is the 4340 steel, ollowed by the glass iber reinorced composite, the carbon iber reinorced composite, 2024-T6 aluminum, and the titanium alloy. Thus, when the issue o economics is considered, there is a signiicant alteration within the ranking scheme. For example, inasmuch as the carbon iber reinorced composite is relatively expensive, it is signiicantly less desirable; in other words, the higher cost o this material may not outweigh the enhanced strength it provides. Table CS1.2 Tabulation o the R/T 2 3 Ratio, Relative Cost ( ), and Product o R/T 2 3 c and c or Five Materials a R/T 2/3 Material {10 2 [Mg (MPa) 2/3 m 3 ]} ($/$) {10 2 ($/$)[Mg/(MPa) 2/3 m 3 ]} 4340 Steel (oil-quenched and tempered) Glass iber reinorced composite (0.65 iber raction) b Carbon iber reinorced composite (0.65 iber raction) b Aluminum alloy (2024-T6) Titanium alloy (Ti-6Al-4V) a The relative cost is the ratio o the price per unit mass o the material and low-carbon steel. c 1R/T 2/3 2 b The ibers in these composites are continuous, aligned, and wound in a helical ashion at a 45 angle relative to the shat axis. c

7 Reerences CS1.7 CS1.3 OTHER PROPERTY CONSIDERATIONS AND THE FINAL DECISION To this point in our materials selection process, we have considered only the strength o materials. Other properties relative to the perormance o the cylindrical shat may be important or example, stiness, and, i the shat rotates, atigue behavior (Sections 8.7 and 8.8 o Introduction; Sections 9.9 and 9.10 o Fundamentals). Furthermore, abrication costs should also be considered; in our analysis, they have been neglected. Relative to stiness, a stiness-to-mass perormance analysis similar to the one just discussed could be conducted. For this case, the stiness perormance index P s is P s 1G (CS1.11) r where G is the shear modulus. The appropriate materials selection chart (log G versus log r) would be used in the preliminary selection process. Subsequently, perormance index and per-unit-mass cost data would be collected on speciic candidate materials; rom these analyses, the materials would be ranked on the basis o stiness perormance and cost. In deciding on the best material, it may be worthwhile to make a table employing the results o the various criteria that were used. The tabulation would include, or all candidate materials, perormance index, cost, and so orth or each criterion, as well as comments relative to any other important considerations. This table puts in perspective the important issues and acilitates the inal decision process. SUMMARY An expression or the strength perormance index was derived or a torsionally stressed cylindrical shat; then, using the appropriate materials selection chart, a preliminary candidate search was conducted. From the results o this search, several candidate engineering materials were ranked on both strength-per-unit mass and cost bases. Other actors that are relevant to the decision-making process were also discussed. REFERENCES Ashby, M. F., CES4 EduPack Cambridge Selector, Granta Design Ltd., Cambridge, UK, Ashby, M. F., Materials Selection in Mechanical Design, 3rd edition, Butterworth-Heinemann, Oxord, Ashby, M. F., and K. Johnson, Materials and Design, Second Edition: The Art and Science o Material Selection in Product Design, Butterworth- Heinemann, Oxord, Ashby, M. F., and D. R. H. Jones, Materials 1: An Introduction to Their Properties and Applications, 3rd edition, Butterworth- Heinemann, Oxord, Ashby, M. F., H. Shercli, and D. Cebon, Materials, Science, Processing and Design, Butterworth-Heinemann, Oxord, Budinski, K. G., and M. K. Budinski, Materials: Properties and Selection, 9th edition, Pearson Prentice Hall, Upper Saddle River, NJ, Dieter, G. E., and L. C. Schmidt, Design: A Materials and Processing Approach, 4th edition, McGraw-Hill, New York, Mangonon, P. L., The Principles o Materials Selection or Design, Pearson Prentice Hall, Upper Saddle River, NJ, 1999.

8 CS1.8 Case Study 1 / Materials Selection or a Torsionally Stressed Cylindrical Shat DESIGN PROBLEMS CS1.D1 (a) Using the procedure outlined in this case study, ascertain which o the metal alloys listed in Appendix B (o Introduction and Fundamentals) have torsional strength perormance indices greater than 10.0 (or and r in units o MPa and g/cm 3, respectively) and, in addition, shear strengths greater than 350 MPa. (b) Also, using the cost database [Appendix C (o Introduction and Fundamentals)], conduct a cost analysis in the same manner as in the case study. For those materials that satisy the criteria noted in part (a), and on the basis o this cost analysis, which material would you select or a solid cylindrical shat? Why? CS1.D2 Perorm a stiness-to-mass perormance analysis on a solid cylindrical shat that is subjected to a torsional stress. Use the same engineering materials that are listed in Table CS1.1. In addition, conduct a material cost analysis. Rank these materials on the basis o both mass o material required and material cost. For glass and carbon iber reinorced composites, assume that the shear moduli are 8.6 and 9.2 GPa, respectively. CS1.D3 (a) A cylindrical cantilever beam is subjected to a orce F, as indicated in the accompanying igure. Derive strength and stiness perormance index expressions analogous to Equations CS1.9 and CS1.11 L r F δ or this beam. The stress imposed on the unixed end s is s FLr (CS1.12) I L, r, and I are, respectively, the length, radius, and moment o inertia o the beam. Furthermore, the beam-end delection d is d FL3 (CS1.13) 3EI where E is the modulus o elasticity o the beam. (b) From the properties database presented in Appendix B (o Introduction and Fundamentals), select the metal alloys with stiness perormance indices greater than 3.0 (or E and r in units o GPa and g/cm 3, respectively). (c) Also, using the cost database [Appendix C (o Introduction and Fundamentals)], conduct a cost analysis in the same manner as the case study. Relative to this analysis and that in part (b), which alloy would you select on a stiness-per-mass basis? (d) Now select the metal alloys with strength perormance indices greater than 14.0 (or s y and r in units o MPa and g/cm 3, respectively), and rank them rom highest to lowest P. (e) Using the cost database, rank the materials in part (d) rom least to most costly. Relative to this analysis and that in part (d), which alloy would you select on a strength-per-mass basis? () Which material would you select i both stiness and strength are to be considered relative to this application? Justiy your choice. CS1.D4 (a) Using the expression developed or stiness perormance index in Problem CS1.D3(a) and data contained in Appendix B (o Introduction and Fundamentals), determine stiness perormance indices or the ollowing polymeric materials: highdensity polyethylene, polypropylene, poly (vinyl chloride), polystyrene, polycarbonate,

9 Design Problems CS1.9 poly(methyl methacrylate), poly(ethylene terephthalate), polytetraluoroethylene, and nylon 6,6. How do these values compare with those o the metallic materials? [Note: In Appendix B (o Introduction and Fundamentals), where ranges o values are given, use average values.] (b) Now, using the cost database [Appendix C (o Introduction and Fundamentals)], conduct a cost analysis in the same manner as the case study. Use cost data or the raw orms o these polymers. (c) Using the expression developed or strength perormance index in Problem CS1.D3(a) and data contained in Appendix B (o Introduction and Fundamentals), determine strength perormance indices or these same polymeric materials. CS1.D5 (a) A bar specimen having a square cross section o edge length c is subjected to a uniaxial tensile orce F, as shown in the accompanying igure. Derive strength and stiness perormance index expressions analogous to Equations CS1.9 and CS1.11 or this bar. F (c) Also, using the cost database [Appendix C (o Introduction and Fundamentals)], conduct a cost analysis in the same manner as the case study. Relative to this analysis and that in part (b), which alloy would you select on a stiness-per-mass basis? (d) Now select the metal alloys with strength perormance indices greater than 120 (or s y and r in units o MPa and g/cm 3, respectively), and rank them rom highest to lowest P. (e) Using the cost database, rank the materials in part (d) rom least to most costly. Relative to this analysis and that in part (d), which alloy would you select on a strength-per-mass basis? () Which material would you select i both stiness and strength are to be considered relative to this application? Justiy your choice. CS1.D6 Consider the plate shown in the accompanying igure, supported at its ends and subjected to a orce F that is uniormly distributed over the upper ace as indicated. The delection d at the L/2 position is given by the expression d 5FL3 32Ewt 3 (CS1.14) w F L t c c L F (b) From the properties database presented in Appendix B (o Introduction and Fundamentals), select the metal alloys with stiness perormance indices greater than 26.0 (or E and r in units o GPa and g/cm 3, respectively). Furthermore, the tensile stress at the underside and also at the L/2 location is equal to s 3FL 4wt 2 (CS1.15)

10 CS1.10 Case Study 1 / Materials Selection or a Torsionally Stressed Cylindrical Shat (a) Derive stiness and strength perormance index expressions analogous to Equations CS1.9 and CS1.11 or this plate. (Hint: Solve or t in these two equations, and then substitute the resulting expressions into the mass equation, as expressed in terms o density and plate dimensions.) (b) From the properties database in Appendix B (o Introduction and Fundamentals), select the metal alloys with stiness perormance indices greater than 1.40 (or E and r in units o GPa and g/cm 3, respectively). (c) Also, using the cost database [Appendix C (o Introduction and Fundamentals)], conduct a cost analysis in the same manner as the case study. Relative to this analysis and that in part (b), which alloy would you select on a stiness-per-mass basis? (d) Now select the metal alloys with strength perormance indices greater than 5.0 (or s y and r in units o MPa and g/cm 3, respectively), and rank them rom highest to lowest P. (e) Using the cost database, rank the materials in part (d) rom least to most costly. Relative to this analysis and that in part (d), which alloy would you select on a strength-per-mass basis? () Which material would you select i both stiness and strength are to be considered relative to this application? Justiy your choice.

REVISED PAGES IMPORTANT TERMS AND CONCEPTS REFERENCES QUESTIONS AND PROBLEMS. 166 Chapter 6 / Mechanical Properties of Metals

REVISED PAGES IMPORTANT TERMS AND CONCEPTS REFERENCES QUESTIONS AND PROBLEMS. 166 Chapter 6 / Mechanical Properties of Metals 1496T_c06_131-173 11/16/05 17:06 Page 166 166 Chapter 6 / Mechanical Properties of Metals IMPORTANT TERMS AND CONCEPTS Anelasticity Design stress Ductility Elastic deformation Elastic recovery Engineering