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DETAILDESIGN

A vast subject. We will concentrate on:

Materials Selection

Process Selection

Cost Breakdown

Materials Selection with Shape

FUNCTION

MATERIAL

PROCESS

SHAPE

SHAPES FOR TENSION,BENDING, TORSION,

BUCKLING--------------------

SHAPE FACTORS--------------------

PERFORMANCE INDICESWITH SHAPE

Common Modes of Loading

Moments of Sections: ElasticSectionShape

A(m2)

I(m4)

K(m4)

2r 4

4r 4

2r

2b 12

4b 414.0 b

ab ba3

4

)( 22

33

baba

bh12

3bh

)(

58.013

3

bhhbhb

2

43 a 332

4a80

34a

rtrr io

2)( 22

tr

rr io

3

44 )(4

tr

rr io

3

44

2

)(2

A = Cross-sectional area

I = Second moment of area

tion ytion

ybyAyIsec

2

sec

2 dd

where y is measured verticallyby is the section width at y

K = Resistance to twisting of section(≡ Polar moment J of a circular section)

tion

rrJsec

3d2G

TLK

where T is the torqueL is the length of the shaftθ is the angle of twistG is the shear modulus

Moments of Sections: Elastic

bt4

tba )(

bthhb io

2)(

)(2 bht

)(2 bht

2

22

41

dt

tb3

32

abta 31

43

2/

)(12

2

33

o

io

bth

hhb

hbth 31

61 3

)4(6

23 btht

8

2dt

43 1

bttb

22

2/5)(4ba

tab

bhhtb

222

bhbt 41

32 3

)8(3

3

hbt

hbht 41

32 3

SectionShape

A(m2)

I(m4)

K(m4)

Moments of Sections: FailureSectionShape

Z(m3)

Q(m3)

3

4r 3

2r

6

3b 321.0 b

ba2

4

)(2

2

ba

ba

6

2bh

)(8.13

22

bhbh

hb

32

3a20

3a

tr

rrr ioo

2

44 )(4

tr

rrr ioo

2

44

2

)(2

Z = Section modulusmyIZ

where

ym is the normal distance from the neutral axis to the outer surface of the beam carrying the highest stress

Q = Factor in twisting similar to Z

TQ

where

is the maximum surface shear stress

Moments of Sections: FailureSectionShape

Z(m3)

Q(m3)

tb2

34

22 12

bttb

abta 31

4

2)(

)(2 2/13

abbat

o

ioo

bth

hhhb

33

6

hbth 31

3

2

tbh2

bhbt 41

32 2

)4(3

23 bthht

)8(3

2

hbt

hbht 41

32 2

4dt

Shape Factors: ElasticBENDING

31

LEICSB

44

24 ArIo

Bending stiffness of a beam

where C1 is a constant depending on the loading details, L is the length of the beam, and E is the Young’s modulus of the material

oBo

BeB I

ISS

Define structure factor as the ratio of the stiffness of the shaped beam to that of a solid circular section with the same cross-sectional area thus:

2

4A

IeB

so,

TORSION

LKGST Torsional stiffness of a beam

where L is the length of the shaft, G is the shear Modulus of the material.

22

24 ArKo 2

2A

KeT

so,

oTo

TeT K

KSS

Define structure factor as the ratio of the torsional stiffness of the shaped shaft to that of a solid circular section with the same cross-sectional area thus:

Shape Factors: Failure/StrengthBENDING

44

2/33 ArZo

ofo

ffB Z

ZMM

Define structure factor as the ratio of the failure moment of the shaped beam to that of a solid circular section with the same cross-sectional area thus:

2/3

4A

ZfB

so,

ZM

IMym

ff ZM

The beam fails when the bending moment is large enough for σ to reach the failure stress of the material:

The highest stress, for a given bending moment M, experienced by a beam is at the surface a distance ym furthest from the neutral axis:

TORSION

QT

The highest shear stress, for a given torque T, experienced by a shaft is given by:

22

2/33 ArQo

2/3

2A

QfT

so,

ofo

ffT Q

QTT

Define structure factor as the ratio of the failure torque of the shaped shaft to that of a solid circular section with the same cross-sectional area thus:

ff QT

The beam fails when the torque is large enough for to reach the failure shear stress of the material:

Shape Factors: Failure/Strength

Please Note:The shape factors for failure/strength described in this lecture course are those defined in the 2nd Edition of “Materials Selection In Mechanical Design” by M.F. Ashby. These shape factors differ from those defined in the 1st Edition of the book. The new failure/strength shape factor definitions are the square root of the old ones.

The shape factors for the elastic case are not altered in the 2nd Edition.

Comparison of Size and Shape

Rectangular sections

I-sections

SIZE →

Shape FactorsSectionShape

Stiffness Failure/StrengtheB e

T fB f

T

11 11

05.13

0.88 0.7418.13

2

ba

ba )( ba

ba

22

2ba

ab

bh

3

)(

58.0132

bhbh

hb

2/1

32

bh

)()/6.01(3

)/(22

2/1

bhhb

hb

0.6221.133

2

73.035

2

2/12

tr

tr

tr 2/12

tr

0.77

Shape Factors cont’dSectionShape

Stiffness Failure/StrengtheB e

T fB f

T

tb

6 4

18

bt

tb

2)/1()/31(

abtaba

222

2/5

))(()(8

babatab

bth

2

2

td

2

2

2)/1(6)/31(

hbthbh

2

32

)/1(6)/41(

hbthbth

3

22

)( bhthb

2)/1(3)/41(

bhbbht

2)/1(6)/81(

hbhhbt

2)/1(3)/41(

hbhhbt

2/1

32

tb 22/1

12

bt

tb

2/3

2/1

)/1()/31(

abab

ta

2/32/1

2/1

)/1(4

bata

2/1)(2

bth

2/3

2/1

)/1()/31(

32

hbhb

th

2/3

322/1

)/1()/41(

2 hbhbt

th

2/1)(

td

2/32/1 )/1()(2

bhbth

2/3

2/1

)/1()/41(

32

hbbh

bt

2/3

2/1

)/1()/81(

18 hbhb

ht

2/3

2/1

)/1()/41(

32

hbhb

ht

Efficiency of Standard Sections

2

4A

IeB

ELASTIC BENDING

Shape Factor:

4

loglog2logeBAI

Rearrange for I and take logs:

Plot logI against logA : parallel lines of slope 2e

B

Efficiency of Standard Sections

2/3

4A

ZfB

BENDING STRENGTH

Shape Factor:

4loglog

23log

fBAZ

Rearrange for I and take logs:

Plot logI against logA : parallel lines of slope 3/2f

B

Efficiency of Standard Sections

ELASTIC TORSION

2

loglog2log;22

eTe

T AKA

K

TORSIONAL STRENGTH

2loglog

23log;2

2/3

fTf

T AQA

Q

N.B. Open sections are good in bending, but poor in torsion

Performance Indices with ShapeELASTIC BENDING

31

LEICSB Bending stiffness of a beam:

ELASTIC TORSION

LKGST Torsional stiffness of a shaft:

2/122/5

2/1

1

4

e

B

B

EL

CSm

f1(F) · f2(G) · f3(M)

2/1

1)( e

BEM So, to minimize mass m, maximise

2

4A

IeB

Shape factor:3

21

4 LAECS

eB

B so,

2/1

2)( e

TEM So, to minimize mass m, maximise

2

2A

KeT

Shape factor:LAGS

eT

T 2

2

so,

2/12

2/32/12

e

TT G

LSm

f1(F) · f2(G) · f3(M)

EG83

Performance Indices with ShapeFAILURE IN BENDING

ff ZM Failure when moment reaches:

FAILURE IN TORSION

ff QT Failure when torque reaches:

3/2

3)( f

BfM So, to minimize mass m, maximise

2/3

4A

ZfB

Shape factor:

4

2/3AMfBf

f so,

3/22/33/2)4(

f

Bff LMm

f1(F) · f2(G) · f3(M)

3/2

4)( f

TfM So, to minimize mass m, maximise

3/22/3

3/24

f

Tff LTm

f1(F) · f2(G) · f3(M)

2/3

2A

QfT

Shape factor:

4

2/3ATf

Tff so,

)2( ff

Shape in Materials Selection Maps

0.01

0.1

1

10

100

1000

0.1 1 10 100

Density, (Mg/m3)

You

ng's

Mod

ulus

, E (G

Pa) Engineering

Alloys

Polymer Foams

Woods

Engineering Polymers

Elastomers

Composites

CeramicsSearch Region

A material with Young’s modulus, E and density, ρ, with a particular section acts as a material with an effective Young’s modulus

and density

eBEE

eB

Performance index for elastic bending including shape,

can be written as

2/1

1)( EM

eB

eB

eBEM

2/1

1)(

EXAMPLE 1, Elastic bending

Φ=1

Φ=10

Shape in Materials Selection Maps

A material with strength, σf and density, ρ, with a particular section acts as a material with an effective strength

and density

2)( fBff

2)( fB

Performance index for failure in bending including shape,

can be written as

3/2

3)( f

fBM

2

3/22

3 )())((

fB

fBfM

EXAMPLE 1, Failure in bending

Engineering Alloys

Polymer Foams

0.1

1

10

100

1000

10000

0.1 1 10 100

Density, (Mg/m3)

Stre

ngth

, f (

MP

a)

Ceramics

Composites

Search Region

Woods

Elastomers

Engineering Polymers

Φ=1

Φ=√10

Micro-Shape FactorsMaterial Micro-Shape

+

Macro-Shape, φ

+

Macro-Shape fromMicro-Shaped Material,

ψφ

=

Up to now we have only considered the role of macroscopic shape on the performance of fully dense materials.

However, materials can have internal shape, “Micro-Shape” which also affects their performance,e.g. cellular solids, foams, honeycombs.

Micro-Shaped Material, ψ

=

Micro-Shaped Material, ψ

Micro-Shape Factors

Prismatic cells

Concentric cylindrical shells with foam between

Fibres embedded in a foam matrix

Consider a solid cylindrical beam expanded, at constant mass, to a circular beam with internal shape (see right).

Stiffness of the solid beam: 31

LIECS oo

Bo

On expanding the beam, its density falls from to , and its radius increases from to

o

oo rr

2/1

or

oo

oo IrrI

24

24

44

The second moment

of area increases to

oo

EE

If the cells, fibres or rings are

parallel to the axis of the beam then

The stiffness of theexpanded beam is thus

oo

B LEIC

LEICS 3

13

1 Shape Factor: o

o

eB S

S

Function

Tie

Beam

Column

Shaft

Mats. Selection: Multiple Constraints

Objective

Minimum cost

Minimum weight

Maximum stored energy

Minimum environmental

impact

Constraint

Stiffness

Strength

Fatigue

GeometryMechanicalThermalElectrical…..

Index

2/1

1EM

Index

3/2

2fM

Materials for Safe Pressure VesselsDESIGN REQUIREMENTS

Function Pressure vessel =contain pressure p

Objective Maximum safety

Constraints (a) Must yield before break(b) Must leak before break(c) Wall thickness small to

reduce mass and cost

Yield before break2

2,

f

ICC

C

IC KCaa

CK

f

ICKM

1

Leak before break

f

IC

ICC

f

KpRCt

CKta

pRtt

pR

22

4

2/2

2,

2

f

ICKM

2

2

Minimum strength

fM 3

Materials for Safe Pressure VesselsSearchRegion

M3 = 100 MPa

M1 = 0.6 m1/2

f

ICKM

1f

ICKM

2

2 fM 3

Material M1

(m1/2)M3

(MPa)Comment

Tough steelsTough Cu alloysTough Al alloys

Ti-alloysHigh strength Al alloysGFRP/CFRP

>0.6>0.6>0.6

0.20.10.1

30012080

700500500

Standard.OFHC Cu.1xxx & 3xxx

High strength, but low safety margin. Good for light vessels.

1. Express the objective as an equation.

2. Eliminate the free variables using each constraint in turn, giving a set of performance equations (objective functions) of the form:

where f, g and h are expressions containing

the functional requirements F, geometry M

and materials indices M.

3. If the first constraint is the most restrictive (known as the active constraint) then the performance is given by P1, and this is maximized by seeking materials with the best values of M1. If the second constraint is the active one then the performance is given by P2 and this is maximized by seeking materials with the best values of M2; and so on.

N.B. For a given Function the Active Constraint will be material dependent.

Multiple Constraints: Formalised

)()()(

)()()()()()(

)()()(

33333

22222

11111

iiiii MhGgFfP

MhGgFfPMhGgFfP

MhGgFfP

Multiple Constraints: A Simple AnalysisA LIGHT, STIFF, STRONG BEAM The object function is ALm

Constraint 1: Stiffness where so,31

LEICS

12

4tI 2/12/5

2/1

11

12E

LCSm B

Constraint 2: Strength where so,Ly

ICFm

ff

22tym 3/2

3/53/2

22

6

f

f LCFm

If the beam is to meet both constraints then, for a given material, its weight is determined by the larger of m1 or m2

or more generally, for i constraints ).....,,max(~321 immmmm

Material E(GPa)

σf

(MPa)

ρ(kgm-3)

m1

(kg)

m2

(kg) (kg)

1020 Steel6061 AlTi 6-4

20570

115

320120950

785027004400

8.75.16.5

16.210.74.4

16.210.76.5

m~Choose a material that minimizes m~

Multiple Constraints: Graphical

log Index M1

log

Inde

x M

2

Construct a materials selection map based on Performance Indices instead of materials properties.

The selection map can be divided into two domains in each of which one constraint is active. The “Coupling Line” separates the domains and is calculated by coupling the Objective Functions:

where CC is the “Coupling Constant”.

222222

11111 )()(

)()( MCMGgFfGgFfM c

Coupling Line M2 = CC·M1

M1 Limited Domain

M2 Limited Domain

A

B

Materials with M2/M1>CC , e.g. , are limited by M1 and constraint 1 is active.

Materials with M2/M1<CC , e.g. , are limited by M2 and constraint 2 is active.

A

B

Multiple Constraints: Graphical

Coupling Line M2 = CC·M1

Search Area

C

log Index M1

log

Inde

x M

2

M1 Limited Domain

M2 Limited Domain

A

B

C

C

A box shaped Search Region is identified with its corner on the Coupling Line.Within this Search Region the performance is maximized whilst simultaneously satisfying both constraints. are good materials.

M1 Limited Domain

M2 Limited Domain

A

B

Coupling Line M2 = CC·M1

log Index M1

log

Inde

x M

2

C

Search Area

A C

Changing the functional requirements F or geometry G changes CC, which shifts the Coupling Line, alters the Search Area, and alters the scope of materials selection.Now and are selectable.

Windings for High Field Magnets

DESIGN REQUIREMENTSFunction Magnet windings

Objective Maximize magnetic field

Constraints (a) No mechanical failure(b) Temperature rise <150°C(c) Radius r and length L of

coil specified

2r dd

LN TurnsCurrent i

B

Upper limits on field and pulse duration are set by the coil material.Field too high the coil fails mechanicallyPulse too long the coil overheats

Classification Pulse Duration

Field Strength

ContinuousLongStandardShortUltra-short

1 s - ∞100 ms-1 s10 - 100 ms10 - 1000 µs0.1 - 10 µs

<30 T30-60 T40-70 T70-80 T>100 T

Windings for High Field Magnets

),(f LNiB foThe field (weber/m2) is

where μo = the permeability of air, N = number of turns, i = current, λf = filling factor,f(α,β) = geometric constant, α = 1+(d/r), β = L/2r

CONSTRAINT 1: Mechanical Failure

Radial pressure created by the field

generates a stress in the coil

),(f2

2

o

Bp

drB

dpr

o ),(f2

2

σ must be less than the yield stress of the coil material σy

and hence 2/1),(f2

rd

B yofailure

So, Bfailure is maximized by maximizing

yM 1

Windings for High Field MagnetsCONSTRAINT 1: Overheating

So, Bheat is maximized by maximizing

e

pCM

2

The energy of the pulse is (Re = average of the resistance over the heating cycle, tpulse = length of the pulse) causes the temperature of the coil to rise by

where Ωe = electrical resistivity of the coil materialCp = specific heat capacity of the coil material

pulseetRi 2

po

pulsee

CdtB

T 22

2

If the upper limit for the change in temperature is ΔTmax and the geometric constant of the coil is included then the second limit on the field is ),(f

2/1

max22

epulse

poheat t

TCdB

Windings for High Field Magnets

Materialσy

(MPa)

ρ(Mg/m3)

Cp

(J/kgK)

Ωe

(10-8Ωm)

Bfailure

(wb/m2)

Bheat

(wb/m2) (wb/m2)

High conductivity CuCu-15%Nb compositeHSLA steel

250780

1600

8.948.907.85

385368450

1.72.425

356289

1139230

356230

Pulse length = 10 ms

B~

),min(~heatfailure BBB In this case the field is limited by the lowest of Bfailure and Bheat: e.g.

),(f2/1

max22

epulse

poheat t

TCdB

2/1),(f2

rd

B yofailure

221 2),(f MCM

tTrdM C

pulse

maxfo

Thus defining the Coupling Line

10

100

1000

10000

100 1000 10000Index M 2 (10-8 m3/J)

Inde

x M

1 (M

Pa)

Search Region:Ultra-short pulse

Search Region:long pulse

Search Region:short pulse

HSLA steels

CuAl-S150.1

Cu-4Sn

Cu-Be-Co-Ni

Be-Coppers

GP coppers

HC Coppers

Cu-NbCu-Al2O3

Cu-Zr

Windings for High Field Magnets

Material Comment

Continuous and long pulseHigh purity coppersPure Silver

Short pulseCu-Al2O3 compositesH-C Cu-Cd alloysH-C Cu-Zr alloysH-C Cu-Cr alloysDrawn Cu-Nb comp’s

Ultra short pulse, ultra high fieldCu-Be-Co-Ni alloysHSLA steels

Best choice for low field, long pulse magnets (heat limited)

Best choice for high field, short pulse magnets (heat and strength limited)

Best choice for high field, short pulse magnets (strength limited)

e

pCM

2yM 1