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Materials and Shape: IFB 2012 Lecture 1 Shape Factors 1/24 Textbook Chapters 9 and 10 Lecture 1 Materials for efficient structures IFB 2012 Materials Selection in Mechanical Design “Efficient” = use least amount of material for given stiffness or strength. Extruded shapes Efficient?

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IFB 2012 Lecture 1 Shape Factors 2/24 Shape and Mechanical Efficiency 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 Is shape important for tie rods?

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IFB 2012 Lecture 1 Shape Factors 3/24 Extruded shapes Certain materials can only be made with certain shapes: what is the best material/shape combination (for each loading mode) ?

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b b Define a standard reference section: a solid square, area A = b 2 IFB 2012 Lecture 1 Shape Factors 4/24 Shape efficiency: bending stiffness pp. 248-249 Area A is constant Area A o = b 2 modulus E unchanged Neutral reference section Shaped sections A o = A Define shape factor for elastic bending, measuring efficiency, as

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IFB 2012 Lecture 1 Shape Factors 5/24 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. bending stiffness

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IFB 2012 Lecture 1 Shape Factors 6/24 I -sections Properties of the shape factor The shape factor is dimensionless -- a pure number. It characterises shape, regardless of size. Circular tubes These sections are φ e times stiffer in bending than a solid square section of the same cross-sectional area Increasing size at constant shape = constant SF Rectangular Sections e = 2

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IFB 2012 Lecture 1 Shape Factors 7/24 Define a standard reference section: a solid square, area A = b 2 Shape efficiency: bending strength p. 254 b b Area A is constant Area A = b 2 yield strength unchanged Neutral reference section Define shape factor for the onset of plasticity (failure), measuring efficiency, as A = A o

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IFB 2012 Lecture 1 Shape Factors 8/24 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. bending strength

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IFB 2012 Lecture 1 Shape Factors 9/24 Tabulation of shape factors (elastic bending) p. 252 A 2 = A o 2 Second moment of section, I

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IFB 2012 Lecture 1 Shape Factors 10/24 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, M f ? 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

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IFB 2012 Lecture 1 Shape Factors 11/24 m = mass A = area L = length = density b = edge length S = stiffness I = second moment of area E = Youngs Modulus Beam (shaped section). Bending stiffness of the beam S: Trick to bring the Shape Factor in ? Eliminating A from the eq. for the mass gives: Chose materials with largest Minimise mass, m, where: Function Objective Constraint L F Area A Shape factor part of the material index Indices that include shape (1): minimise mass at constant stiffness allowing for changes in shape p. 265-270 L F

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IFB 2012 Lecture 1 Shape Factors 12/24 Indices that include shape (2): minimise mass at constant strength p. 311 m = mass A = area L = length = density M f = bending strength I = moment of section E = Young’s Modulus Z = section modulus Beam (shaped section). Bending strength of the beam M f : Trick to bring the Shape Factor in ? Eliminating A from the equation for m gives: Chose materials with largest Minimise mass, m, where: Function Objective Constraint Shape factor part of the material index L F Area A L F

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IFB 2012 Lecture 1 Shape Factors 13/24 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. Given shape, Material 1, given S Same shape, Material 2, same S

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IFB 2012 Lecture 1 Shape Factors 14/24 Demystifying Shape Factors (elastic bending) Shaping (material fixed) at constant bending stiffness reduces the mass of the component in proportion to e -1/2. Optimum approach: simultaneously maximise both M and . Square beam, m o, given S Shaped to φ e, same material, same S L F L F EXAM QUESTION Is the cross section area constant when going from m o to m s ?

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IFB 2012 Lecture 1 Shape Factors 15/24 Demystifying Shape Factors (failure of beams) Square beam, m o, M f Shaped to φ f, same material, same M f 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 . L F L F EXAM QUESTION: Is the cross section area constant when going from m o to m s ?

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IFB 2012 Lecture 1 Shape Factors 16/24 Material , Mg/m 3 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 4 11.4 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. L F

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IFB 2012 Lecture 1 Shape Factors 17/24 Note that new material with Tute #3: p.269-270 Al: e = 44 Al: e = 1 Density (Mg/m 3 ) Young’s modulus (GPa) Material substitution at constant structural stiffness allowing for differences in cross sectional shape/size to increase the structural efficiency We call this “dragging the material’s label”

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IFB 2012 Lecture 1 Shape Factors 18/24 Dragging the material labels in CES shaping at constant stiffness Drag the labels along lines of slope 1 Selection line of slope 2 Unshaped Steel SF =1 Unshaped Aluminium 0 Unshaped Bamboo SF= 1 Shaped aluminium SF = 44 Shaping makes Steel beams competitive with Al beams and Bamboo cane Shaped Bamboo SF=5.6 Shaped steel SF=65

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IFB 2012 Lecture 1 Shape Factors 19/24 Note that new material with Dragging the material’s label in CES shaping at constant strength Material substitution at constant structural strength allowing for differences in cross sectional shape/size to increase the structural efficiency

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IFB 2012 Lecture 1 Shape Factors 20/24 Dragging the material labels in CES shaping at constant strength Selection line of slope 1.5 Shaped Steel SF=7; (SF) 2 =49 Shaped Bamboo SF=2 (SF) 2 =4 Shaping makes Steel beams competitive with Al beams and Bamboo cane Shaped Aluminium SF=10; (SF) 2 =100

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Steel, Al and Bamboo scaffoldings IFB 2012 Lecture 1 Shape Factors 21/24 Shaping allows you to choose. Use what is more mass-efficient, convenient, cheap, and, of course, available.

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IFB 2012 Lecture 1 Shape Factors 22/24 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/S o = I/I o f = M/M o = Z/Z o 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/ e 1/2 (stiffness) or by 1/ f 2/3 (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. Exam questions:

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Really scary bamboo scaffoldings IFB 2012 Lecture 1 Shape Factors 23/24 http://www.flickr.com/photos/photocapy/41857542/

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IFB 2012 Lecture 1 Shape Factors 24/24 -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.)

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IFB 2012 Lecture 1 Shape Factors 25/24 End of Lecture 1 IFB 2012

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IFB 2012 Lecture 1 Shape Factors 26/24 Shape and mode of loading Standard structural members Loading: tension/compression Both, Area A and shape I XX, I YY matter Both, Area A and shape J matter Both, Area A and shape I min matter Area A matters, not shape Loading: bending Loading: torsion Loading: axial compression

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IFB 2012 Lecture 1 Shape Factors 27/24 Examples of Materials Indices including shape p. 278 Buckling: Same as elastic bending

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IFB 2012 Lecture 1 Shape Factors 28/24 Shape factors for twisting and buckling p. 252/253 Failure under torsion Buckling Same as elastic bending Elastic twisting

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