Materials Selection for Mechanical Design II · PDF fileMaterials Selection for Mechanical Design II ... Chart from the CES EduPack 2005, Granta Design Limited, Cambridge, UK. (c)

Massachusetts Institute of TechnologyCambridge, Massachusetts Materials Systems Laboratory

©Jeremy Gregory and Randolph Kirchain, 2005 Materials Selection II – Slide 1

Materials Selection Materials Selection for for

Mechanical Design IIMechanical Design II

A Brief Overview of a Systematic A Brief Overview of a Systematic MethodologyMethodology

Material Material andand Shape SelectionShape Selection


©Jeremy Gregory and Randolph Kirchain, 2005 Materials Selection – Slide 2

Method for Early Technology ScreeningMethod for Early Technology ScreeningDesign performance is Design performance is determined by the determined by the combination of:combination of:

ShapeShapeMaterialsMaterialsProcessProcess

Underlying principles of Underlying principles of selection are unchangedselection are unchanged

BUT, do not underestimate BUT, do not underestimate impact of shape or the impact of shape or the limitation of processlimitation of process

Materials

Process

Shape



Material and Shape SelectionMaterial and Shape SelectionPerformance isn't just about materials Performance isn't just about materials -- shape can shape can also play an important rolealso play an important roleShape can be optimized to maximize performance Shape can be optimized to maximize performance for a given loading conditionfor a given loading conditionSimple crossSimple cross--sectional geometries are not always sectional geometries are not always optimal optimal

Efficient Shapes like IEfficient Shapes like I--beams, tubes can be betterbeams, tubes can be betterShape is limited by materialShape is limited by material

Wood can be formed only so thinWood can be formed only so thinGoal is to optimize both shape and material for a Goal is to optimize both shape and material for a given loading conditiongiven loading condition



Loading Conditions and Shape

Different loading Different loading conditions are conditions are enhanced by enhanced by maximizing different maximizing different geometric propertiesgeometric propertiesArea for tensionArea for tensionSecond moment for Second moment for compression and compression and bendingbendingPolar moment for Polar moment for torsiontorsion

Figure by MIT OCW.

T

FF

T

Bending : Beam

Twisting : Shaft

Compression : Column

FF

Area A δ

δ

θ

Area A moment I

r

Area A polar moment J

Area A moment I

F

Tension : TieFAσ =

MyIσ =

TrJτ =

nπ2 EIl2Fcrit =



Area

SecondMoment

PolarMoment

2roh

b 2ro

h

b

t

b

t

h

Shapes and MomentsShapes and Moments

bh 2rπ ( )2 2o ir rπ − ( )2t h b+ ( )2t h b+

3

12bh 4

4rπ ( )4 4

4 o ir rπ− 31 1 3

6bh th

⎛ ⎞+⎜ ⎟⎝ ⎠

31 1 36

bh th

⎛ ⎞+⎜ ⎟⎝ ⎠

3

1 0.583bh b

h⎛ ⎞−⎜ ⎟⎝ ⎠

4

2rπ ( )4 4

2 o ir rπ− ( )

42 22 1tb h th b h

⎛ ⎞−⎜ ⎟+ ⎝ ⎠32 1 4

3hbtb

⎛ ⎞+⎜ ⎟⎝ ⎠

2ri



Shape Factor DefinitionShape Factor DefinitionShape factor measures efficiency for a Shape factor measures efficiency for a mode of loading given an equivalent crossmode of loading given an equivalent cross--sectionsection

““EfficiencyEfficiency””: For a given loading condition, : For a given loading condition, section uses as little material as possiblesection uses as little material as possible

Defined as 1 for a solid crossDefined as 1 for a solid cross--sectionsectionHigher number is better, more efficientHigher number is better, more efficiente

o

SS

φ =For elastic cases:For elastic cases:φ φ = shape factor= shape factorS S = stiffness of cross= stiffness of cross--section under questionsection under questionSSo o = stiffness of reference solid cross= stiffness of reference solid cross--sectionsection



Shape Factor for Elastic BendingShape Factor for Elastic Bending

eB

o o o

S EI IS EI I

φ = = =

4 2

12 12o o

ob AI = =bo

bo

Reference solid crossReference solid cross--sectionsection

2

12eB

o

I II A

φ = = Notice that shape factor is Notice that shape factor is dimensionlessdimensionless

Compare sections Compare sections of same area of same area ⇒⇒

oA A=



II--Beam Elastic Bending Shape FactorBeam Elastic Bending Shape Factor

bo

bo b

t

h

2

11

o o

o

o

A bbA

===

( )

3

2

211 1 3 1.125612 13.5

o

eB

A t h bA A

bI h th

IA

φ

= +

= =

⎛ ⎞= + =⎜ ⎟⎝ ⎠

= =

tt = 0.125= 0.125hh = 3= 3bb = 1= 1

For these dimensions, the shape For these dimensions, the shape increased stiffness increased stiffness over 13 timesover 13 times while while using the same amount of material!using the same amount of material!

Is this design possible in all materials?Is this design possible in all materials?



Materials Limit Best Achievable Shape Materials Limit Best Achievable Shape FactorFactor

Shape efficiency dependent on materialShape efficiency dependent on materialConstraints: manufacturing, material properties, local bucklingConstraints: manufacturing, material properties, local buckling

For example, canFor example, can’’t have thin sections of woodt have thin sections of woodValues in table determined empiricallyValues in table determined empiricallyNote: previous design not possible in polymers, wood (Note: previous design not possible in polymers, wood (φeB)=13.5)=13.5

MaterialStructural Steels 65 25Aluminum Alloys 44 31

GFRP and CFRP 39 26

Polymers 12 8Woods 6 1

Bending Torsion

( )max

eBφ ( )

max

eTφ



Shape Factors and Material IndicesShape Factors and Material IndicesExample: Bending BeamExample: Bending Beam

( )

3

2

23

1/ 25/ 2

1/ 2

12

12

12

eB

o

e eB B

eB

m ALF CEIS

LI II A

C EI S AL

SA m LC E

ρ

δ

φ

φ φ

ρ

φ

=

= ≥

= =

=

⎡ ⎤⎛ ⎞ ⎢ ⎥= ⎜ ⎟ ⎢ ⎥⎝ ⎠

⎣ ⎦

Mass:

Bending Stiffenss:

Shape Factor:

Replace in Stiffness using :

Eliminate from mass using stiffness:

Material Ind( )1/ 2 1/ 2eBE EM M

φ

ρ ρ= =ex: Previously:



Shape Factors and Material Indices: BeamsShape Factors and Material Indices: Beams

((φφffTTσσff))22/3/3//ρρ((φφee

TTGG))1/21/2//ρρTorsionTorsion((φφff

BBσσff))22/3/3//ρρ((φφeeBBEE))1/21/2//ρρBendingBending

σσff//ρρE/E/ρρTensionTension

Strength LimitedStrength LimitedStiffness Stiffness LimitedLimitedLoadingLoading

Objective: Objective: Minimize MassMinimize Mass

Performance Metric:Performance Metric: MassMass

Maximize!Maximize!



Shape Factors Affect Material ChoiceShape Factors Affect Material ChoiceShape factors can Shape factors can dramatically improve dramatically improve performance for a performance for a given loading given loading conditionconditionThe optimal The optimal combination of shape combination of shape and material leads to and material leads to the best designthe best design

Elastomers

PolymersWoods

Composites

Metals

Ceramics

Foams

M with φ=1

M with φ=10



Example Problem: Bicycle ForksExample Problem: Bicycle Forks

Bicycle forks need to be lightweightBicycle forks need to be lightweightPrimary constraint can be stiffness or Primary constraint can be stiffness or strengthstrengthToughness and cost can be other Toughness and cost can be other constraintsconstraints

Photos of bicycle forks removed for copyright reasons.



Bicycle Forks: Problem DefinitionBicycle Forks: Problem DefinitionFunction:Function:

Forks Forks -- support support bending loadsbending loads

Objective:Objective:Minimize massMinimize mass

Constraints:Constraints:Length specifiedLength specifiedMust not fail Must not fail (strength constraint)(strength constraint)

Free variables:Free variables:Material Material Area: Tube radius OR Area: Tube radius OR thickness OR shapethickness OR shape

LL

FF 42

3

3/ 2

42

4

Objective:

Constraint:

Free Variables:

Solid Tube:

Hollow Tube:

Shape:

m mf

fB

m

m ALMy FLyI I

rA r I

A rt I r t

Z IZA y

ρ

σ σ

ππ

π π

πφ

=

= = ≤

= =

≈ ≈

= =



Material Indices: Shape specifiedMaterial Indices: Shape specifiedFree variable definition importantFree variable definition important

( )

3

1/3

2/31/3 2/32/3

2/3

4

4

4

Solid SectionFree Variable: Area

Solve for :

Substitute into :

Maximize:

mf

f

f

f

f

MyIFLr

r

FLr

m

m F L

M

σ σ

σπ

πσ

ρπσ

σρ

= ≤

≥

⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠

⎡ ⎤= ⎢ ⎥

⎢ ⎥⎣ ⎦⎡ ⎤

= ⎢ ⎥⎢ ⎥⎣ ⎦

( ) ( )

2

1/ 2

1/ 21/ 2 31/ 2

1/ 2

4

Hollow SectionFree Variable: Radius

Solve for :

Substitute into :

Maximize:

mf

f

f

f

f

MyIFLr t

r

FLrt

m

m F L t

M

σ σ

σπ

π σ

ρπσ

σρ

= ≤

≥

⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠

⎡ ⎤= ⎢ ⎥

⎢ ⎥⎣ ⎦⎡ ⎤

= ⎢ ⎥⎢ ⎥⎣ ⎦

2

2

2

:

2

Hollow SectionFree Variable: Thickness

Solve for

Substitute into :

Maximize:

mf

f

f

f

f

MyIFLr t

tFLtr

m

Lm Fr

M

σ σ

σπ

π σ

ρσ

σρ

= ≤

≥

=

⎡ ⎤= ⎢ ⎥

⎢ ⎥⎣ ⎦⎡ ⎤

= ⎢ ⎥⎣ ⎦



Material Index with Shape FreeMaterial Index with Shape Free

( )( )

( )

2/3 5/32/3

2 /3

4

Substitute into :

Maximize:

fB f

fB f

m

m F L

M

ρπφ σ

φ σ

ρ

⎡ ⎤⎢ ⎥=⎢ ⎥⎣ ⎦

⎡ ⎤⎢ ⎥=⎢ ⎥⎣ ⎦

3/ 2

2 /3

4

4

Solve for :

mf f

B

fB f

FLy FL FLI Z AA

FLA

πσφ

πφ σ

≥ = =

⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠



Material indices with shape factors change Material indices with shape factors change material selectionmaterial selection

919135354.254.251.51.5375375CFRPCFRP

444417174.254.251.81.8165165Magnesium AZ 61Magnesium AZ 61

727222225.95.94.424.42955955Titanium 6Titanium 6--44

484815155.95.92.72.7250250AluAlu (6061(6061--T6)T6)

454512127.57.57.827.82880880Steel (Reynolds 531)Steel (Reynolds 531)

595935352.22.20.70.7120120BambooBamboo

36363636110.510.518080Spruce (Norwegian)Spruce (Norwegian)

((φφffBBσσff))2/32/3//ρρσσff

2/32/3//ρρφφffBBρρ (Mg/m3)(Mg/m3)σσff ((MPaMPa))MaterialMaterial

*Material Index w/out shape factor*Material Index w/out shape factor **Material Index **Material Index withwith shape factorshape factor

** ****



Strength ConstraintStrength Constraint

Density (kg/m^3)100 1000 10000

Tens

ile S

treng

th (P

a)

1e6

1e7

1e8

1e9

Bamboo

Softwood: pine, along grain

Hardwood: oak, along grain

Wrought magnesium alloys

CFRP, epoxy matrix (isotropic)

Age-hardening wrought Al-alloysTitanium alloys

Medium carbon steel

Chart from the CES EduPack 2005, Granta Design Limited, Cambridge, UK. (c) Granta Design. Courtesy of Granta Design Limited. Used with permission.__________

http://grantadesign.com



Stiffness ConstraintStiffness Constraint

Density (kg/m^3)100 1000 10000

You

ng's

Mod

ulus

(Pa)

1e6

1e7

1e8

1e9

1e10

1e11

Age-hardening wrought Al-alloysMedium carbon steel

Bamboo

Softwood: pine, along grain

CFRP, epoxy matrix (isotropic)

Hardwood: oak, along grain

Wrought magnesium alloys

Chart from the CES EduPack 2005, Granta Design Limited, Cambridge, UK. (c) Granta Design. Courtesy of Granta Design Limited. Used with permission.__________

http://grantadesign.com



Material for floor joists

Wood beam Steel I-beam

Density (g/cm3) ~0.58 ~7.9

Modulus (GPa) ~10 ~210

Material Cost ($/kg) ~$0.90 ~$0.65

2.0-2.2 15-25

E1/2/Cmρ ~6.1 ~2.8

~8.8 ~12.6

Example of Material Selection including Example of Material Selection including Shape: Floor JoistsShape: Floor Joists

φeB

(φeBE)1/2/Cmρ

*Material Index w/out shape factor*Material Index w/out shape factor **Material Index **Material Index withwith shape factorshape factor

**

****

Documents

Materials Selection for Mechanical Design II · PDF fileMaterials Selection for Mechanical Design II ... Chart from the CES EduPack 2005, Granta Design Limited, Cambridge, UK. (c)