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 IFB 2012 Lecture 1 Shape Factors

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

IFB 2012 Lecture 1 Shape Factors 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

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

IFB 2012 Lecture 1 Shape Factors 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

Properties of the shape factor The shape factor is dimensionless -- a pure number. It characterises shape, regardless of size. Rectangular Sections e = 2 I-sections Circular tubes Increasing size at constant shape = constant SF 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

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

IFB 2012 Lecture 1 Shape Factors 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

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

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

IFB 2012 Lecture 1 Shape Factors Indices that include shape (1): minimise mass at constant stiffness allowing for changes in shape p. 265-270 L F Beam (shaped section). L F 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: Chose materials with largest IFB 2012 Lecture 1 Shape Factors

IFB 2012 Lecture 1 Shape Factors Indices that include shape (2): minimise mass at constant strength p . 311 L F Beam (shaped section). L F 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: Chose materials with largest Shape factor part of the material index IFB 2012 Lecture 1 Shape Factors

From Introduction: Demystifying Material Indices (elastic bending) Given shape, Material 1, given S Same shape, Material 2, same S 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

Demystifying Shape Factors (elastic bending) Shaped to φe, same material, same S Square beam, mo , given S L F EXAM QUESTION Is the cross section area constant when going from mo to ms? 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 . IFB 2012 Lecture 1 Shape Factors

Demystifying Shape Factors (failure of beams) Square beam, mo , Mf Shaped to φf, same material, same Mf L F 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

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  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 4 11.4 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

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

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

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

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

IFB 2012 Lecture 1 Shape Factors 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

IFB 2012 Lecture 1 Shape Factors 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

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

IFB 2012 Lecture 1 Shape Factors -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

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

Shape and mode of loading Standard structural members Loading: tension/compression Area A matters, not shape Loading: bending Both, Area A and shape IXX, IYY matter 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

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

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