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2 - Ashby Method
2.3 - Materials Selection for
best performance
Outline
Resources:
• M. F. Ashby, “Materials Selection in Mechanical Design” Butterworth Heinemann, 1999
Chapters 5 and 6
• The Cambridge Material Selector (CES) software -- Granta Design, Cambridge
(www.grantadesign.com)
• Deriving performance indices
• Performance maximizing criteria
• Selection with multiple constraints
Materials selection and function
� Design involves choosing a material, process and part shape to perform some function.
� Function dictates the choice of both materials and shape.
� In many cases materials choice is not directly dependent of shape.
Performance Indices
Minimum cost
Minimum
weight
Maximum energy storage
Minimum environ. impact
FUNCTION
OBJECTIVE
CONSTRAINTS
INDEX
Tie
Beam
Shaft
Column
Mechanical,Thermal,
Electrical...
Stiffness
specified
Strengthspecified
Fatigue limit
Geometry
specified
=
yσ
ρM
Minimise
this!
Each combination of
Function
ObjectiveConstraintFree variable
Has a
characterising material index
Performance Indices
Deriving Performance Indices: Procedure
� Identify the attribute to be maximized or minimized (weight, cost, energy, stiffness, strength, safety,
environmental damage, etc.).
� Develop an equation for this attribute in terms of the functional requirements, the geometry and the material properties (the objective function).
� Identify the free (unspecified) variables.
� Identify the constraints; rank them in order of importance.
� Develop equations for the constraints (no yield; no fracture; no buckling, maximum heat capacity, cost below
target, etc.).
� Substitute for the free variables from the constraints into
the objective function.
� Group the variables into three groups: functional requirements, F, geometry, G, and material properties, M,
thus, we can write:
p = f1(F) f2 (G) f3 (M)
� Read off the performance index, expressed as a quantity f3 (M), to be minimized or maximized.
Deriving Performance Indices: Procedure
Deriving Performance Indices: Light, strong tie
Strong tie of length L and minimum mass
L
FF
Area A
m = massA = area
L = length
ρ = density= yield strength
yσ
Deriving Performance Indices: Light, strong tie
• Minimize mass, m , of a solid cylindrical tie rod of length L, which
carries a tensile force F with safety factor Sf. The mass is given by:
m = A L ρ
where A is the area of the cross-section and ρ is the density. This is called the Objective Function
• The length L and force F are specified; the radius r is free.
• The section must, however, be sufficient to carry the tensile load F,
requiring that:
F / A = σf / Sf
where σf is the failure strength.
Deriving Performance Indices: Light, strong tie
• Eliminating A between these two equations gives:
m = ( Sf F ) ( L ) { ρ / σf } or
• Note the form of this result.
• The first bracket contains the functional requirementthat the specified load is safely supported.
• The second bracket contains the specified geometry(the length of the tie).
• The last bracket contains the material properties.
σ
ρ=
y
FLm
Deriving Performance Indices: Light, strong tie
• We want to minimize the performance, m , while meeting the
functional and geometric requirements.
This means we want the smallest value of { ρ / σf } or the largest value of
M = σf / ρ
Ashby defines this latter quantity as the performance index.
Tie-rod
Minimise mass m
Strong tie of length L and minimum mass
L
FF
Area A
• Length L is specified
• Must not fail under load F
• Adequate fracture toughness
• Material choice
• Section area A
Function
Objective
Constraints
Free variables
m = mass
A = areaL = length
ρ = density= yield strength
yσ
STEP 1
Identify function, constraints,
objective and free variables
Deriving Performance Indices: Light, strong tie
Tie-rod
Minimise mass m:
m = A L ρρρρ (1)
Strong tie of length L and minimum mass
L
FF
Area A
• Length L is specified
• Must not fail under load F
• Adequate fracture toughness
• Material choice
• Section area A
Function
Objective
Constraints
Free variables
m = mass
A = areaL = length
ρ = density= yield strength
yσ
STEP 2
Define equation for objective -- the “performance equation”
Deriving Performance Indices: Light, strong tie
Tie-rod
Minimise mass m:
m = A L ρρρρ (1)
Strong tie of length L and minimum mass
L
FF
Area A
• Length L is specified
• Must not fail under load F
• Adequate fracture toughness
• Material choice
• Section area A
Function
Objective
Constraints
Free variables
m = mass
A = areaL = length
ρ = density= yield strength
yσ
STEP 3
If the “performance equation”contains a free variable other than material, identify the
constraint that limits it
Equation for constraint on A:
F/A < σσσσy (2)
Deriving Performance Indices: Light, strong tie
Tie-rod
Minimise mass m:
m = A L ρρρρ (1)
Strong tie of length L and minimum mass
L
FF
Area A
• Length L is specified
• Must not fail under load F
• Adequate fracture toughness
Function
Objective
Constraints
Free variables
m = mass
A = areaL = length
ρ = density= yield strength
yσ
Equation for constraint on A:
F/A < σσσσy (2)
STEP 4
Use this constraint to
eliminate the free variable in performance equation
• Material choice• Section area A;
eliminate in (1) using (2):
σ
ρ=
y
FLm
Deriving Performance Indices: Light, strong tie
Tie-rod
Minimise mass m:
m = A L ρρρρ (1)
Strong tie of length L and minimum mass
L
FF
Area A
• Length L is specified
• Must not fail under load F
• Adequate fracture toughness
Function
Objective
Constraints
Free variables
m = mass
A = areaL = length
ρ = density= yield strength
yσ
Equation for constraint on A:
F/A < σσσσy (2) STEP 5
Read off the combination of material properties that maximise performance
• Material choice• Section area A;
eliminate in (1) using (2):
σ
ρ=
y
FLm
Deriving Performance Indices: Light, strong tie
Tie-rod
Minimise mass m:
m = A L ρρρρ (1)
Strong tie of length L and minimum mass
L
FF
Area A
• Length L is specified
• Must not fail under load F
• Adequate fracture toughness
Function
Objective
Constraints
Free variables
m = mass
A = areaL = length
ρ = density= yield strength
yσ
Equation for constraint on A:
F/A < σσσσy (2)
PERFORMANCEINDEX
• Material choice• Section area A;
eliminate in (1) using (2):
σ
ρ=
y
FLm Chose materials with smallest
yσρ
Deriving Performance Indices: Light, strong tie
Deriving Performance Indices: Light, stiff tie
Tie-rod
Minimise mass m:
m = A L ρρρρ (1)
=
E
ρSLm 2
Stiff tie of length L and minimum mass
L
FF
Area A
• Material choice
• Section area A; eliminate in (1) using (2):
Function
Objective
Constraints
Free variables
Chose materials with smallest
Eρ
m = mass
A = areaL = lengthρ = densityS = stiffness
E = Youngs Modulus
Stiffness of the tie S:
L
AES = (2)
Performance Indices for weight: Tie
Function Stiffness Strength
Tension (tie)
Bending (beam)
Bending (panel)
Cost, Cm
Density, ρ
Modulus, E
Strength, σy
Endurance limit, σe
Thermal conductivity, λ
T- expansion coefficient, α
the “Physicists” view of materials, e.g.
Material properties --
the “Engineers” view of materials
Material indices --
ρ/E yρ/σ
1/2ρ/E
2/3
yρ/σ
Objective: minimise mass
Minimise these!
1/3ρ/E
1/2y
ρ/σ
Deriving Performance Indices: Light, stiff beam
m = massA = area
L = length
ρ = densityb = edge lengthS = stiffness
I = second moment of areaE = Youngs Modulus
Beam (solid square section).
Stiffness of the beam S:
I is the second moment of area:
• Material choice.• Edge length b. Combining the equations gives:
3L
IECS =
12
bI
4
=
ρ
=
2/1
2/15
EC
LS12m
ρ=ρ= LbLAm 2
Chose materials with smallest
ρ2/1E
b
b
L
F
Minimise mass, m, where:
ρ
Function
Objective
Constraint
Free variables
Deriving Performance Indices: Light, strong beam
m = massA = areaL = length
ρ = densityb = edge lengthI = second moment of area
σy = yield strength
Beam (solid square section).
Must not fail under load F
I is the second moment of area:
• Material choice.• Edge length b. Combining the equations gives:
=
⋅>
3y b
3FL
I
b/2Mσ
12
bI
4
=
( ) ( )
=
2/3
y
2/35/3
σ
ρ3FLm
ρ=ρ= LbLAm 2
Chose materials with smallest
2/3
yσ
ρ
b
b
L
F
Minimise mass, m, where:
Function
Objective
Constraint
Free variables
Performance Indices for weight: Beam
Function Stiffness Strength
Tension (tie)
Bending (beam)
Bending (panel)
Cost, Cm
Density, ρ
Modulus, E
Strength, σy
Endurance limit, σe
Thermal conductivity, λ
T- expansion coefficient, α
the “Physicists” view of materials, e.g.
Material properties --
the “Engineers” view of materials
Material indices --
ρ/E yρ/σ
1/2ρ/E
2/3
yρ/σ
Objective: minimise mass
Minimise these!
1/3ρ/E
1/2y
ρ/σ
Deriving Performance Indices: Light, stiff panel
m = massw = widthL = lengthρ = densityt = thicknessS = stiffnessI = second moment of areaE = Youngs Modulus
Panel with given width w and length L
Stiffness of the panel S:
I is the second moment of area:
3L
IECS =
12
twI
3
=
tw
ρ
=
3/12
3/12
EL
C
wS12m
L
F
ρ=ρ= LtwLAm
Chose materials with smallest
ρ3/1E
• Material choice.• Panel thickness t. Combining the equations gives:
Minimise mass, m, where
Function
Objective
Constraint
Free variables
Deriving Performance Indices: Light, strong panel
m = massw = widthL = lengthρ = densityt = thicknessI = second moment of area
σy = yield strength
Panel with given width w and length L
Must not fail under load F
I is the second moment of area:
12
twI
3
=
tw
( ) ( )
= 1/2
y
3/21/2
σ
ρL3Fwm
L
F
ρ=ρ= LtwLAm
• Material choice.• Panel thickness t. Combining the equations gives:
Minimise mass, m, where
Function
Objective
Constraint
Free variables
=
⋅>
2y wt
3FL
I
t/2Mσ
Chose materials with smallest
1/2
yσ
ρ
Performance Indices for weight: Panel
Function Stiffness Strength
Tension (tie)
Bending (beam)
Bending (panel)
Cost, Cm
Density, ρ
Modulus, E
Strength, σy
Endurance limit, σe
Thermal conductivity, λ
T- expansion coefficient, α
the “Physicists” view of materials, e.g.
Material properties --
the “Engineers” view of materials
Material indices --
ρ/E yρ/σ
1/2ρ/E 2/3
yρ/σ
Objective: minimise mass
Minimise these!
1/3ρ/E 1/2
yρ/σ
Minimum cost
Minimum
weight
Maximum energy storage
Minimum environ. impact
FUNCTION
OBJECTIVE
CONSTRAINTS
INDEX
Tie
Beam
Shaft
Column
Mechanical,Thermal,
Electrical...
Stiffness
specified
Strengthspecified
Fatigue limit
Geometry
specified
ρ=
2/1EM
Minimise
this!
Optimised selection using charts
b
b
L
F
Optimised selection using charts
Index 1/2E
ρM =
22 M/ρE =
( ) ( ) ( )MLog2Log2ELog −ρ=
Contours of constantM are lines of slope 2
on an E-ρ chart
CE 2/1
=ρ
0.1
10
1
100
Metals
Polymers
Elastomers
Woods
Composites
Foams0.01
1000
1000.1 1 10Density (Mg/m3)
Young’s
modulu
s E
, (G
Pa)
Ceramics
2
Optimised selection using charts
Index 1/2E
ρM =
22 M/ρE =
( ) ( ) ( )MLog2Log2ELog −ρ=
0.1
10
1
100
Metals
Polymers
Elastomers
Woods
Composites
Foams
1000
1000.1 1 10
Density (Mg/m3)
Young’s
modulu
s E
, (G
Pa)
Ceramics
2
y = a x + b
ρ = Log(ρ) = -1 0 1
Contours of constantM are lines of slope 2
on an E-ρ chart
y = Log (E)
x = Log (ρ)
Optimised selection using charts
Index 1/2E
ρM =
22 M/ρE =
( ) ( ) ( )MLog2Log2ELog −ρ=
0.1
10
1
100
Metals
Polymers
Elastomers
Woods
Composites
Foams
1000
1000.1 1 10
Density (Mg/m3)
Young’s
modulu
s E
, (G
Pa)
Ceramics
2
y = a x + b
ρ =
b = 1-2Log(M) =1M = 10-1/2 = 0.31
y = Log (E)
x = Log (ρ)
Log(ρ) = -1 0 1
Optimised selection using charts
Index 1/2E
ρM =
22 M/ρE =
( ) ( ) ( )MLog2Log2ELog −ρ=
0.1
10
1
100
Metals
Polymers
Elastomers
Woods
Composites
Foams
1000
1000.1 1 10
Density (Mg/m3)
Young’s
modulu
s E
, (G
Pa)
Ceramics
2y = a x + b
ρ =
b = 2-2Log(M) =2M = 10-1 = 0.1
Minimising M
y = Log (E)
x = Log (ρ)
Log(ρ) = -1 0 1
Minimum cost
Minimum
weight
Maximum energy storage
Minimum environ. impact
FUNCTION
OBJECTIVE
CONSTRAINTS
INDEX Stiffness
specified
Strengthspecified
Fatigue limit
Geometry
specified
( )[ ]EfM ,ρ=
Minimise
this!
Each combination ofFunction
ObjectiveConstraintFree variable
Has a characterising material index
Performance Indices for weight: Stiffness
Performance Indices for weight: Stiffness
Function Stiffness Strength
Tension (tie)
Bending (beam)
Bending (panel)
Cost, Cm
Density, ρ
Modulus, E
Strength, σy
Endurance limit, σe
Thermal conductivity, λ
T- expansion coefficient, α
the “Physicists” view of materials, e.g.
Material properties --
the “Engineers” view of materials
Material indices --
ρ/E yρ/σ
1/2ρ/E 2/3
yρ/σ
Objective: minimise mass
Minimise these!
1/3ρ/E 1/2
yρ/σ
Performance Indices for weight: Stiffness
Index 1/2E
ρM =
22 M/ρE =
( ) ( ) ( )MLog2Log2ELog −ρ=
Contours of constantM are lines of slope 2
on an E-ρ chart
CE
=ρ
CE 2/1
=ρ
CE 3/1
=ρ
0.1
10
1
100
Metals
Polymers
Elastomers
Woods
Composites
Foams0.01
1000
1000.1 1 10Density (Mg/m3)
Young’s
modulu
s E
, (G
Pa)
Ceramics
1
2 3
Performance Indices for weight
Deriving Performance Indices for Cost and Energy
� To minimize Cost use the indices for minimum weight, replacing density ρρρρ by Cρρρρ, where C is the cost per kg.
� To minimize Energy use the indices for minimum weight, replacing density ρρρρ by qρρρρ, where q is the energy content per kg.
M = ρ / σf M = Cρ / σf
M = ρ / σf M = qρ / σf
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