1. 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

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:
IFB 2012 Lecture 1 Shape Factors

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

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

10. 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

11. IFB 2012 Lecture 1 Shape Factors
Indices that include shape (1): minimise mass at constant stiffness allowing for changes in shape p 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

12. 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

13. 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

14. 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

15. 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

16. 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/m E, GPa e,max
1020 Steel
6061 Al
GFRP
Wood (oak)
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. We call this “dragging the material’s label”
Tute #3: p
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)

18. 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

19. 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

20. 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

21. 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

22. 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

23. Really scary bamboo scaffoldings
IFB 2012 Lecture 1 Shape Factors

24. 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

25. IFB 2012 Lecture 1 Shape Factors
End of Lecture 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
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

27. 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

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