2 - Ashby Method - · PDF file2 - Ashby Method 2.3 - Materials Selection for best performance Outline Resources: ... where A is the area of the cross-section and ρ is the density

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

Performance Indices

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


• 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.


• 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


• 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


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


yσ

STEP 1

Identify function, constraints,

objective and free variables


Tie-rod

Minimise mass m:

m = A L ρρρρ (1)


L

FF

Area A




• Material choice

• Section area A

Function

Objective

Constraints

Free variables

m = mass

A = areaL = length


yσ

STEP 2

Define equation for objective -- the “performance equation”


Tie-rod

Minimise mass m:



L

FF

Area A




• Material choice

• Section area A

Function

Objective

Constraints

Free variables

m = mass

A = areaL = length


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)


Tie-rod

Minimise mass m:



L

FF

Area A




Function

Objective

Constraints

Free variables

m = mass

A = areaL = length


yσ


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


Tie-rod

Minimise mass m:



L

FF

Area A




Function

Objective

Constraints

Free variables

m = mass

A = areaL = length


yσ


F/A < σσσσy (2) STEP 5

Read off the combination of material properties that maximise performance



σ

ρ=

y

FLm


Tie-rod

Minimise mass m:



L

FF

Area A




Function

Objective

Constraints

Free variables

m = mass

A = areaL = length


yσ


F/A < σσσσy (2)

PERFORMANCEINDEX



σ

ρ=

y

FLm Chose materials with smallest

yσρ


Deriving Performance Indices: Light, stiff tie

Tie-rod

Minimise mass m:


=

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


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


• 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


2/3

yσ

ρ

b

b

L

F

Minimise mass, m, where:

Function

Objective

Constraint

Free variables

Performance Indices for weight: Beam


Tension (tie)

Bending (beam)

Bending (panel)

Cost, Cm

Density, ρ

Modulus, E

Strength, σy







Material indices --

ρ/E yρ/σ

1/2ρ/E

2/3

yρ/σ


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:


3L

IECS =

12

twI

3

=

tw

ρ

=

3/12

3/12

EL

C

wS12m

L

F

ρ=ρ= LtwLAm


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


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


1/2

yσ

ρ

Performance Indices for weight: Panel


Tension (tie)

Bending (beam)

Bending (panel)

Cost, Cm

Density, ρ

Modulus, E

Strength, σy







Material indices --

ρ/E yρ/σ

1/2ρ/E 2/3

yρ/σ


Minimise these!

1/3ρ/E 1/2

yρ/σ

Minimum cost

Minimum

weight



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


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


Index 1/2E

ρM =

22 M/ρE =


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


on an E-ρ chart

y = Log (E)

x = Log (ρ)


Index 1/2E

ρM =

22 M/ρE =


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


Index 1/2E

ρM =

22 M/ρE =


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



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



Tension (tie)

Bending (beam)

Bending (panel)

Cost, Cm

Density, ρ

Modulus, E

Strength, σy







Material indices --

ρ/E yρ/σ

1/2ρ/E 2/3

yρ/σ


Minimise these!

1/3ρ/E 1/2

yρ/σ


Index 1/2E

ρM =

22 M/ρE =



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

Documents

2 - Ashby Method - · PDF file2 - Ashby Method 2.3 - Materials Selection for best performance Outline Resources: ... where A is the area of the cross-section and ρ is the density