Materials Selection for Mechanical Design II A Brief Overview of a Systematic
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Materials Selection for Mechanical Design II A Brief Overview of a Systematic Methodology Material and Shape Selection Massachusetts Institute of Technology Cambridge, Massachusetts ©Jeremy Gregory and Randolph Kirchain, 2005 Materials Systems Laboratory Materials Selection II – Slide 1 Method for Early Technology Screening Design performance is determined by the combination of: Shape Materials Process Underlying principles of selection are unchanged Materials Process Shape BUT, do not underestimate impact of shape or the limitation of process Massachusetts Institute of Technology Cambridge, Massachusetts ©Jeremy Gregory and Randolph Kirchain, 2005 Materials Systems Laboratory Materials Selection – Slide 2 Material and Shape Selection Performance isn't just about materials - shape can also play an important role Shape can be optimized to maximize performance for a given loading condition Simple cross-sectional geometries are not always optimal Shape is limited by material Efficient Shapes like I-beams, tubes can be better Wood can be formed only so thin Goal is to optimize both shape and material for a given loading condition Massachusetts Institute of Technology Cambridge, Massachusetts ©Jeremy Gregory and Randolph Kirchain, 2005 Materials Systems Laboratory Materials Selection – Slide 3 Loading Conditions and Shape Different loading conditions are enhanced by maximizing different geometric properties Area for tension Second moment for compression and bending Polar moment for torsion σ= F A F F δ Area A Tension : Tie F σ= My I δ Area A moment I τ= Tr J r θ Area A polar moment J Fcrit = nπ2 EI l2 Bending : Beam T T Twisting : Shaft F Area A moment I F Compression : Column Figure by MIT OCW. Massachusetts Institute of Technology Cambridge, Massachusetts ©Jeremy Gregory and Randolph Kirchain, 2005 Materials Systems Laboratory Materials Selection – Slide 4 Shapes and Moments 2ro h 2ri h 2ro b Area bh πr Second Moment bh3 12 π Polar Moment bh3 ⎛ b⎞ − 1 0.58 ⎜ ⎟ 3 ⎝ h⎠ Massachusetts Institute of Technology Cambridge, Massachusetts ©Jeremy Gregory and Randolph Kirchain, 2005 4 π 2 2 r4 r t t 4 π ( ro2 − ri 2 ) π r ( 4 π 4 o r ( 2 4 o b b 2t ( h + b ) 2t ( h + b ) 4 ) 1 3 ⎛ b⎞ 1 3 ⎛ b⎞ h t ⎜1 + 3 ⎟ h t ⎜1 + 3 ⎟ h⎠ 6 ⎝ h⎠ 6 ⎝ 4 ) 2tb 2 h 2 ⎛ t ⎞ ⎜1 − ⎟ (h + b) ⎝ h ⎠ − ri − ri h 4 h⎞ 2 3⎛ bt ⎜ 1 + 4 ⎟ b⎠ 3 ⎝ Materials Systems Laboratory Materials Selection – Slide 5 Shape Factor Definition Shape factor measures efficiency for a mode of loading given an equivalent crosssection “Efficiency”: For a given loading condition, section uses as little material as possible Defined as 1 for a solid cross-section Higher number is better, more efficient S e φ = So Massachusetts Institute of Technology Cambridge, Massachusetts ©Jeremy Gregory and Randolph Kirchain, 2005 For elastic cases: φ = shape factor S = stiffness of cross-section under question So = stiffness of reference solid cross-section Materials Systems Laboratory Materials Selection – Slide 6 Shape Factor for Elastic Bending S EI I φ = = = So EI o I o e B Reference solid cross-section 4 2 bo Ao Io = = 12 12 bo bo I 12 I φ = = 2 Io A e B Massachusetts Institute of Technology Cambridge, Massachusetts ©Jeremy Gregory and Randolph Kirchain, 2005 Compare sections of same area ⇒ Ao = A Notice that shape factor is dimensionless Materials Systems Laboratory Materials Selection – Slide 7 I-Beam Elastic Bending Shape Factor t bo t = 0.125 h=3 b=1 h bo Ao = bo2 bo = 1 Ao = 1 For these dimensions, the shape increased stiffness over 13 times while using the same amount of material! Is this design possible in all materials? Massachusetts Institute of Technology Cambridge, Massachusetts ©Jeremy Gregory and Randolph Kirchain, 2005 b A = 2t ( h + b ) A = 1 = Ao 1 3 ⎛ b⎞ I = h t ⎜ 1 + 3 ⎟ = 1.125 6 ⎝ h⎠ 12 I e φB = 2 = 13.5 A Materials Systems Laboratory Materials Selection – Slide 8 Materials Limit Best Achievable Shape Factor Shape efficiency dependent on material Constraints: manufacturing, material properties, local buckling For example, can’t have thin sections of wood Values in table determined empirically Note: previous design not possible in polymers, wood (φeB)=13.5 Bending Material Structural Steels Aluminum Alloys (φ ) e B max Torsion (φ ) e T max 65 44 25 31 GFRP and CFRP 39 26 Polymers Woods 12 6 8 1 Massachusetts Institute of Technology Cambridge, Massachusetts ©Jeremy Gregory and Randolph Kirchain, 2005 Materials Systems Laboratory Materials Selection – Slide 9 Shape Factors and Material Indices Example: Bending Beam Mass: m = AL ρ Bending Stiffenss: S = Shape Factor: φBe = F ≥ δ CEI L3 I 12 I = 2 Io A Replace I in Stiffness using φBe : S = C E e 2 φ A 3 B 12 L 1/ 2 ⎛ 12 S ⎞ Eliminate A from mass using stiffness: m = ⎜ ⎟ ⎝ C ⎠ (φBe E ) ⎡ ρ L5 / 2 ⎢ ⎢ φeE ⎣ B ( 1/ 2 Material Index: M = Massachusetts Institute of Technology Cambridge, Massachusetts ©Jeremy Gregory and Randolph Kirchain, 2005 ρ Previously: M = ⎤ ⎥ 1/ 2 ⎥ ⎦ ) E1/ 2 ρ Materials Systems Laboratory Materials Selection – Slide 10 Shape Factors and Material Indices: Beams Objective: Minimize Mass Performance Metric: Mass Tension Stiffness Limited E/ρ Bending (φeBE)1/2/ρ (φfBσf)2/3/ρ Torsion (φeTG)1/2/ρ (φfTσf)2/3/ρ Loading Strength Limited σf/ρ Maximize! Massachusetts Institute of Technology Cambridge, Massachusetts ©Jeremy Gregory and Randolph Kirchain, 2005 Materials Systems Laboratory Materials Selection – Slide 11 Shape Factors Affect Material Choice Shape factors can dramatically improve performance for a given loading condition The optimal combination of shape and material leads to the best design Massachusetts Institute of Technology Cambridge, Massachusetts ©Jeremy Gregory and Randolph Kirchain, 2005 Ceramics Composites M with φ=10 Woods Foams M with φ=1 Metals Polymers Elastomers Materials Systems Laboratory Materials Selection – Slide 12 Example Problem: Bicycle Forks Photos of bicycle forks removed for copyright reasons. Bicycle forks need to be lightweight Primary constraint can be stiffness or strength Toughness and cost can be other constraints Massachusetts Institute of Technology Cambridge, Massachusetts ©Jeremy Gregory and Randolph Kirchain, 2005 Materials Systems Laboratory Materials Selection – Slide 13 Bicycle Forks: Problem Definition Function: Objective: Minimize mass L Constraints: Forks - support bending loads Length specified Must not fail (strength constraint) Free variables: Material Area: Tube radius OR thickness OR shape Massachusetts Institute of Technology Cambridge, Massachusetts ©Jeremy Gregory and Randolph Kirchain, 2005 F Objective: m = AL ρ Mym FLym Constraint: σ = = ≤σ f I I Free Variables: Solid Tube: A = π r 2 I= π r4 4 Hollow Tube: A ≈ 2π rt I ≈ π r 3t 4 πZ Shape: φ = 3/ 2 A f B I Z= ym Materials Systems Laboratory Materials Selection – Slide 14 Material Indices: Shape specified Free variable definition important Solid Section Hollow Section Hollow Section Free Variable: Area Free Variable: Radius Free Variable: Thickness Mym ≤σ f I 4 FL σf ≥ 3 πr Solve for r: σ= Mym ≤σ f I FL σf ≥ 2 πr t Solve for r: Mym ≤σ f I FL σf ≥ 2 πr t Solve for t : σ= 1/ 3 ⎛ 4 FL ⎞ r =⎜ ⎜ πσ f ⎟⎟ ⎝ ⎠ Substitute into m: ⎡ ρ ⎤ 2/3 m = π 1/ 3 ( 4 F ) L2 / 3 ⎢ 2 / 3 ⎥ ⎢⎣ σ f ⎥⎦ ⎡ σ 2f / 3 ⎤ Maximize: M = ⎢ ⎥ ⎢⎣ ρ ⎥⎦ Massachusetts Institute of Technology Cambridge, Massachusetts ©Jeremy Gregory and Randolph Kirchain, 2005 σ= 1/ 2 FL π r 2σ f ⎛ FL ⎞ r =⎜ ⎜ π tσ f ⎟⎟ ⎝ ⎠ Substitute into m: t= ⎡ ρ ⎤ 1/ 2 m = ( 4π F ) L3t ⎢ 1/ 2 ⎥ ⎢⎣ σ f ⎥⎦ ⎡ σ 1/f 2 ⎤ Maximize: M = ⎢ ⎥ ⎢⎣ ρ ⎥⎦ L2 m = 2F r ( ) 1/ 2 Substitute into m: ⎡ ρ ⎢ ⎢⎣ σ f ⎤ ⎥ ⎥⎦ ⎡σ f ⎤ Maximize: M = ⎢ ⎥ ⎣ ρ ⎦ Materials Systems Laboratory Materials Selection – Slide 15 Material Index with Shape Free FLym FL FL 4 π σf ≥ = = f 3/ 2 φB A I Z Solve for A: ⎛ FL 4 π A=⎜ f ⎜ φB σ f ⎝ ⎞ ⎟⎟ ⎠ 2/3 Massachusetts Institute of Technology Cambridge, Massachusetts ©Jeremy Gregory and Randolph Kirchain, 2005 Substitute into m: ⎡ ⎤ ρ 5/ 3 ⎢ ⎥ m= 4 πF L ⎢ φ f σ 2/3 ⎥ ⎣ B f ⎦ ⎡ φ f σ 2/3 ⎤ B f ⎢ ⎥ Maximize: M = ⎢ ⎥ ρ ⎣ ⎦ ( ) 2/3 ( ( ) ) Materials Systems Laboratory Materials Selection – Slide 16 Material indices with shape factors change material selection * ** σf (MPa) ρ (Mg/m3) φ fB σf2/3/ρ (φfBσf)2/3/ρ Spruce (Norwegian) 80 0.51 1 36 36 Bamboo 120 0.7 2.2 35 59 Steel (Reynolds 531) 880 7.82 7.5 12 45 Alu (6061-T6) 250 2.7 5.9 15 48 Titanium 6-4 955 4.42 5.9 22 72 Magnesium AZ 61 165 1.8 4.25 17 44 CFRP 375 1.5 4.25 35 91 Material *Material Index w/out shape factor Massachusetts Institute of Technology Cambridge, Massachusetts ©Jeremy Gregory and Randolph Kirchain, 2005 **Material Index with shape factor Materials Systems Laboratory Materials Selection – Slide 17 Strength Constraint Medium carbon steel Age-hardening w rought Al-alloys Titanium alloys 1e9 CFRP, epoxy m atrix (isotropic) Tensile Strength (Pa) Wrought m agnesium alloys Hardw ood: oak, along grain 1e8 Softw ood: pine, along grain Bam boo 1e7 1e6 100 1000 10000 Density (kg/m^3) Chart from the CES EduPack 2005, Granta Design Limited, Cambridge, UK. (c) __________ Granta Design. Courtesy of Granta Design Limited. Used with permission. Massachusetts Institute of Technology Cambridge, Massachusetts ©Jeremy Gregory and Randolph Kirchain, 2005 Materials Systems Laboratory Materials Selection – Slide 18 Stiffness Constraint Medium carbon steel Age-hardening w rought Al-alloys CFRP, epoxy m atrix (isotropic) 1e11 Wrought m agnesium alloys Hardw ood: oak, along grain Bam boo Young's Modulus (Pa) 1e10 Softw ood: pine, along grain 1e9 1e8 1e7 1e6 100 1000 10000 Density (kg/m^3) Chart from the CES EduPack 2005, Granta Design Limited, Cambridge, UK. (c) __________ Granta Design. Courtesy of Granta Design Limited. Used with permission. Massachusetts Institute of Technology Cambridge, Massachusetts ©Jeremy Gregory and Randolph Kirchain, 2005 Materials Systems Laboratory Materials Selection – Slide 19 Example of Material Selection including Shape: Floor Joists Wood beam Steel I-beam Material for floor joists * ** Density (g/cm3) ~0.58 ~7.9 Modulus (GPa) ~10 ~210 Material Cost ($/kg) ~$0.90 ~$0.65 φeB 2.0-2.2 15-25 E1/2/Cmρ ~6.1 ~2.8 (φeBE)1/2/Cmρ ~8.8 ~12.6 *Material Index w/out shape factor Massachusetts Institute of Technology Cambridge, Massachusetts ©Jeremy Gregory and Randolph Kirchain, 2005 **Material Index with shape factor Materials Systems Laboratory Materials Selection – Slide 20
