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Materials and Shape:

IFB 2012 Materials Selection in Mechanical Design. Efficient?. Lecture 1. Materials and Shape:. “Efficient” = use least amount of material for given stiffness or strength. Materials for efficient structures. Extruded shapes. Textbook Chapters 9 and 10.

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Materials and Shape:

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  1. IFB 2012Materials Selection in Mechanical Design Efficient? Lecture 1 Materials and Shape: “Efficient” = use least amount of material for given stiffness or strength. Materials for efficient structures Extruded shapes Textbook Chapters 9 and 10 IFB 2012 Lecture 1 Shape Factors

  2. Shape and Mechanical Efficiency Is shape important for tie rods? • Section shape becomes important when materials are loaded in bending, in torsion, or are used as slender columns. • Examples of “Shape”: • Shapes to which a material can be formed are limited by the material itself. Shapes from: http://www.efunda.com/math/areas/RolledSteelBeamsS.cfm IFB 2012 Lecture 1 Shape Factors

  3. Certain materials can only be made with certain shapes: what is the best material/shape combination (for each loading mode) ? Extruded shapes IFB 2012 Lecture 1 Shape Factors

  4. Neutral reference section Shaped sections Area A is constant b b Shape efficiency: bending stiffness pp. 248-249 Define a standard reference section: a solid square, area A = b2 Defineshape factor for elastic bending, measuring efficiency, as Area Ao = b2 modulus E unchanged Ao = A IFB 2012 Lecture 1 Shape Factors

  5. bending stiffness A shaped beam of shape factor for elastic bending, e = 10,is 10 times stiffer than a solid square section beam of similar cross section area. IFB 2012 Lecture 1 Shape Factors

  6. Increasing size at constant shape = constant SF Properties of the shape factor • The shape factor is dimensionless -- a pure number. • It characterises shape, regardless of size. e = 2 Rectangular Sections I-sections Circular tubes • These sections are φe times stiffer in bending than a solid square section of the same cross-sectional area IFB 2012 Lecture 1 Shape Factors

  7. Neutral reference section Area A is constant yield strength unchanged b b Shape efficiency: bending strength p. 254 Define a standard reference section: a solid square, area A = b2 Defineshape factor for the onset of plasticity (failure), measuring efficiency, as Area A = b2 A = Ao IFB 2012 Lecture 1 Shape Factors

  8. bending strength A shaped beam of shape factor for bending strength, f = 10,is 10 times stronger than a solid square section beam of similar cross section area. IFB 2012 Lecture 1 Shape Factors

  9. A2 = Ao2 Second moment of section, I Tabulation of shape factors (elastic bending) p. 252 IFB 2012 Lecture 1 Shape Factors

  10. Comparison of shapes done so far for given material (E, y) and constant cross section area, A.Interesting, but not very useful. • How to compare different materials and different shapes at: • Constant structural stiffness, S ? • Constant failure moment, Mf ? This is a case of Material Substitution at constant structural stiffness or strength, allowing for differences in shape Example: compare Steel scaffoldings with Bamboo scaffoldings IFB 2012 Lecture 1 Shape Factors

  11. F L F L Chose materials with largest Indices that include shape (1): minimise mass at constant stiffness allowing for changes in shape p. 265-270 Beam (shaped section). Function Area A Objective Minimise mass, m, where: Constraint Bending stiffness of the beam S: m = mass A = area L = length  = density b = edge length S = stiffness I = second moment of area E = Youngs Modulus Trick to bring the Shape Factor in ? Shape factor part of the material index Eliminating A from the eq. for the mass gives: IFB 2012 Lecture 1 Shape Factors

  12. F L F L Chose materials with largest Indices that include shape (2): minimise mass at constant strength p . 311 Beam (shaped section). Function Area A Objective Minimise mass, m, where: Constraint Bending strength of the beam Mf: m = mass A = area L = length  = density Mf = bending strength I = moment of section E = Young’s Modulus Z = section modulus Trick to bring the Shape Factor in ? Eliminating A from the equation for m gives: Shape factor part of the material index IFB 2012 Lecture 1 Shape Factors

  13. Given shape, Material 1, given S Same shape, Material 2, same S From Introduction: Demystifying Material Indices (elastic bending) For given shape, the reduction in mass at constant bending stiffness is determined by the reciprocal of the ratio of material indices. Same conclusion applies to bending strength. IFB 2012 Lecture 1 Shape Factors

  14. Shaped to φe, same material, same S F Square beam, mo, given S L F L Demystifying Shape Factors (elastic bending) EXAM QUESTION Is the cross section area constant when going from mo to ms? Shaping (material fixed) at constant bending stiffnessreduces the mass of the component in proportion to e-1/2. Optimum approach: simultaneously maximise both M and . IFB 2012 Lecture 1 Shape Factors

  15. F Shaped to φf, same material, same Mf Square beam, mo , Mf L F L Demystifying Shape Factors (failure of beams) EXAM QUESTION: Is the cross section area constant when going from mo to ms? Shaping (material fixed) at constant bending strength reduces the mass of the component in proportion to f-2/3. Optimum approach: simultaneously maximise both M and . IFB 2012 Lecture 1 Shape Factors

  16. Material , Mg/m3 E, GPae,max 1020 Steel 7.85 205 65 1.8 14.7 6061 Al 2.70 70 44 3.1 20.4 GFRP 1.75 28 39 2.9 18.9 Wood (oak) 0.9 13 8 411.4 F L Practical examples of material-shape combinations • Materials for stiff beams of minimum weight • Fixed shape (e fixed): choose materials with greatest • Shape e a variable: choose materials with greatest  Same shape for all (up to e = 8): wood is best Maximum shape factor (e = e,max): Al-alloy is best Steel recovers some performance through high e,max See textbook pp. 266 and 268 for more examples. IFB 2012 Lecture 1 Shape Factors

  17. Al: e = 1 Al: e = 44 Tute #3: p.269-270 Note that new material with We call this “dragging the material’s label” Young’s modulus (GPa) Material substitution at constant structural stiffnessallowing for differences in cross sectional shape/size to increase the structural efficiency IFB 2012 Lecture 1 Shape Factors Density (Mg/m3)

  18. Selection line of slope 2 Unshaped Aluminium0 Unshaped Steel SF =1 Unshaped Bamboo SF= 1 Shaped aluminium SF = 44 Shaped Bamboo SF=5.6 Shaped steel SF=65 Dragging the material labels in CES  shaping at constant stiffness Drag the labels along lines of slope 1 Shaping makes Steel beams competitive with Al beams and Bamboo cane IFB 2012 Lecture 1 Shape Factors

  19. Note that new material with Dragging the material’s label in CES  shaping at constant strength Material substitution at constant structural strengthallowing for differences in cross sectional shape/size to increase the structural efficiency IFB 2012 Lecture 1 Shape Factors

  20. Selection line of slope 1.5 Shaped Steel SF=7; (SF)2=49 Shaped Bamboo SF=2 (SF)2=4 Shaped Aluminium SF=10; (SF)2=100 Dragging the material labels in CES  shaping at constant strength Shaping makes Steel beams competitive with Al beams and Bamboo cane IFB 2012 Lecture 1 Shape Factors

  21. Steel, Al and Bamboo scaffoldings • Shaping allows you to choose. Use what is more mass-efficient, convenient, cheap, and, of course, available. IFB 2012 Lecture 1 Shape Factors

  22. Exam questions: Shaping at constant cross section A increases the bending stiffness or strength by  at constant mass. This stems from the definition of shape factor e = S/So= I/Io f = M/Mo = Z/Zo Dragging the material label in the CES charts is equivalent to shaping at constant bending stiffness or strength, so the mass is reduced by 1/e1/2 (stiffness) or by 1/f2/3 (strength). Dragging the material label along a line of slope 1 keeps the ratio E/ρ = E*/ρ* constant (* = shaped). Shaping sacrifices the stiffness in tension (tie rod) in favour of the bending stiffness (beam), thus increasing the mass efficiency of the section. IFB 2012 Lecture 1 Shape Factors

  23. Really scary bamboo scaffoldings http://www.flickr.com/photos/photocapy/41857542/ IFB 2012 Lecture 1 Shape Factors

  24. -Tutorial 1, Materials and Shape. • Solve in this order: (4 Exercises) • E9.1 (p. 623) • CASE STUDY 10.2 (p.279) • CASE STUDY 10.4 (p. 284) • E9.8 (p. 627) Notes and Hints for E9.1 and CS10.4: E9.1 does not require the use of charts. CS 10.4: follow the procedure of case study 10.2; create a CES chart and analyse the effect of shaping on the position of the bubbles (Do that by dragging the materials’ labels.) IFB 2012 Lecture 1 Shape Factors

  25. End of Lecture 1 IFB 2012 IFB 2012 Lecture 1 Shape Factors

  26. Shape and mode of loading Standard structural members Loading: tension/compression Area A matters, not shape Both, Area A and shape IXX, IYY matter Loading: bending Both, Area A and shape J matter Loading: torsion Both, Area A and shape Imin matter Loading: axial compression IFB 2012 Lecture 1 Shape Factors

  27. Examples of Materials Indices including shape p. 278 Buckling: Same as elastic bending IFB 2012 Lecture 1 Shape Factors

  28. Shape factors for twisting and buckling p. 252/253 Elastic twisting Failure under torsion Buckling Same as elastic bending IFB 2012 Lecture 1 Shape Factors

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