MECH Lecture 6 (1/3) Shape Factors 1/23 Material and Shape: Textbook Chapters 11 and 12 Lecture 6 (1/3) Efficient? Materials for efficient structures Tute # 3, 6 exercises, 3 afternoons to solve them. Read the instructions (BB) carefully. Due Mo Sept. 15, 4:00 pm MECH Materials Selection in Mechanical Design Laboratory Group II: Meet on Friday Au 29, 8 am, room 413, Bldng 43. “Efficient” = use least amount of material for given stiffness or strength. To create a deformation work-stress chart for foams, use (densification strain * yield strength) as y-axis and yield strength as x-axis.) See Announcement in Bb.

MECH Lecture 6 (1/3) Shape Factors 2/23 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: Is shape important for tie rods?

MECH Lecture 6 (1/3) Shape Factors 3/23 Shape and mode of loading Standard structural members Loading: tension/compression Area A and shape I XX, I YY matter Area A and shape J matter Area A and shape I min matter Area A matters, not shape Loading: bending Loading: torsion Loading: axial compression Certain materials can be made to certain shapes: what is the best material/shape combination (for each loading mode) ?

MECH Lecture 6 (1/3) Shape Factors 4/23 Shape efficiency: bending stiffness pp b b Area A is constant Area A o = b 2 modulus E unchanged Neutral reference section Shaped sections Define a standard reference section: a solid square, area A = b 2 Moments of Sections; p 477 A o = A Define shape factor for elastic bending, measuring efficiency, as

MECH Lecture 6 (1/3) Shape Factors 5/23 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

MECH Lecture 6 (1/3) Shape Factors 6/23 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

MECH Lecture 6 (1/3) Shape Factors 7/23 Define a standard reference section: a solid square, area A = b 2 Shape efficiency: bending strength p. 294/5 b b Area A is constant Area A = b 2 yield strength unchanged Neutral reference section Moments of Sections; p 477 Define shape factor for the onset of plasticity (failure), measuring efficiency, as A = A o

MECH Lecture 6 (1/3) Shape Factors 8/23 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

MECH Lecture 6 (1/3) Shape Factors 9/23 Tabulation of shape factors (elastic bending) p. 292/3 A 2 = A o 2 Second moment of section, I

MECH Lecture 6 (1/3) Shape Factors 10/23 Comparison of shapes done so far at constant material (E,  y ) and given cross section area, A How to compare different materials and different shapes at: Constant structural stiffness, S ? Constant failure moment, M f ? Material substitution at constant stiffness or strength allowing for differences in shape

MECH Lecture 6 (1/3) Shape Factors 11/23 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 p. 310

MECH Lecture 6 (1/3) Shape Factors 12/23 Indices that include shape (2): minimise mass at constant strength p. 311 m = mass A = area L = length  = density M f = bending strength I = second moment of area E = Youngs 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 L F Area A Shape factor part of the material index

MECH Lecture 6 (1/3) Shape Factors 13/23 From Lecture 4: Demystifying Material Indices (elastic bending) For given shape, the reduction in mass at constant bending stiffness is determined by the ratio of material indices. Same conclusion applies to bending strength. Unshaped mass, Material 1 Unshaped mass Material 2

MECH Lecture 6 (1/3) Shape Factors 14/23 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 . Unshaped mass Shaped mass, same material, same S Q: Is the cross section area constant when going from m o to m s ?

MECH Lecture 6 (1/3) Shape Factors 15/23 Demystifying Shape Factors (failure of beams) Unshaped mass Shaped mass, 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 . EXAM QUESTION: Is the cross section area constant when going from m o to m s ?

MECH Lecture 6 (1/3) Shape Factors 16/23 Material , Mg/m 3 E, GPa  e,max 1020 Steel Al GFRP Wood (oak) 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

MECH Lecture 6 (1/3) Shape Factors 17/23 Note that new material with Shape on selection charts: stiffness p. 312/3 Al:  e = 44 Al:  e = 1 Density (Mg/m 3 ) Young’s modulus (GPa) Material substitution at constant stiffness or strength allowing for differences in shape

MECH Lecture 6 (1/3) Shape Factors 18/23 Shape on selection charts: stiffness p. 314 Drag the labels along lines of slope 1 Selection line of slope 2 Unshaped Steel SF =1 Unshaped Aluminium Unshaped Bamboo SF= 1 Shaped aluminium SF = 44 Shaping makes Steel competitive with Al and Bamboo Shaped Bamboo SF=5.6 Shaped steel SF=65

MECH Lecture 6 (1/3) Shape Factors 19/23 Note that new material with Shape on selection charts: strength p. 314 Material substitution at constant stiffness or strength allowing for differences in shape

MECH Lecture 6 (1/3) Shape Factors 20/23 Shape on selection charts: strength p. 314 Selection line of slope 1.5 Shaped Steel SF=7; (SF) 2 =49 Shaped Bamboo SF=2 (SF) 2 =4 Shaping makes Steel competitive with Al and Bamboo Shaped Aluminium SF=10; (SF) 2 =100

MECH Lecture 6 (1/3) Shape Factors 21/23 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 labels 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). Exam question: (to get everybody confused!)

MECH Lecture 6 (1/3) Shape Factors 22/23 Examples of indices including shape p. 318 Same as elastic bending

MECH Lecture 6 (1/3) Shape Factors 23/23 This afternoon: solve Exercises E8.1, 8.8, 8.9 and Leave 8.6 / 8.7 for next sessions. -Tutorial 3 (E8. Materials and Shape) (6 Exercises). Solve in this order: E8.1; E8.8; E8.9; E8.12; (solve either E8.6 or E8.7) (see hints and instructions in BB). Exercise #6 for Tute 3: Show that the shape factors of Table 12.5 (p. 325) are a factor 4/3 = 1.33 too large.

MECH Lecture 6 (1/3) Shape Factors 24/23 Example using CES: dragging labels

MECH Lecture 6 (1/3) Shape Factors 25/23 End of Lecture 6 two more lectures re. shape factors to follow

MECH Lecture 6 (1/3) Shape Factors 26/23 Shape factors for twisting and buckling Failure under torsion p. 296 Buckling p. 296 Same as elastic bending Elastic twisting p. 294 Moments of Sections; p 477