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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
Massachusetts Institute of TechnologyCambridge, Massachusetts Materials Systems Laboratory
©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
Massachusetts Institute of TechnologyCambridge, Massachusetts Materials Systems Laboratory
©Jeremy Gregory and Randolph Kirchain, 2005 Materials Selection – Slide 3
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
Massachusetts Institute of TechnologyCambridge, Massachusetts Materials Systems Laboratory
©Jeremy Gregory and Randolph Kirchain, 2005 Materials Selection – Slide 4
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 =
Massachusetts Institute of TechnologyCambridge, Massachusetts Materials Systems Laboratory
©Jeremy Gregory and Randolph Kirchain, 2005 Materials Selection – Slide 5
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
Massachusetts Institute of TechnologyCambridge, Massachusetts Materials Systems Laboratory
©Jeremy Gregory and Randolph Kirchain, 2005 Materials Selection – Slide 6
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
Massachusetts Institute of TechnologyCambridge, Massachusetts Materials Systems Laboratory
©Jeremy Gregory and Randolph Kirchain, 2005 Materials Selection – Slide 7
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=
Massachusetts Institute of TechnologyCambridge, Massachusetts Materials Systems Laboratory
©Jeremy Gregory and Randolph Kirchain, 2005 Materials Selection – Slide 8
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?
Massachusetts Institute of TechnologyCambridge, Massachusetts Materials Systems Laboratory
©Jeremy Gregory and Randolph Kirchain, 2005 Materials Selection – Slide 9
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
Massachusetts Institute of TechnologyCambridge, Massachusetts Materials Systems Laboratory
©Jeremy Gregory and Randolph Kirchain, 2005 Materials Selection – Slide 10
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:
Massachusetts Institute of TechnologyCambridge, Massachusetts Materials Systems Laboratory
©Jeremy Gregory and Randolph Kirchain, 2005 Materials Selection – Slide 11
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!
Massachusetts Institute of TechnologyCambridge, Massachusetts Materials Systems Laboratory
©Jeremy Gregory and Randolph Kirchain, 2005 Materials Selection – Slide 12
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
Massachusetts Institute of TechnologyCambridge, Massachusetts Materials Systems Laboratory
©Jeremy Gregory and Randolph Kirchain, 2005 Materials Selection – Slide 13
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.
Massachusetts Institute of TechnologyCambridge, Massachusetts Materials Systems Laboratory
©Jeremy Gregory and Randolph Kirchain, 2005 Materials Selection – Slide 14
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
=
= = ≤
= =
≈ ≈
= =
Massachusetts Institute of TechnologyCambridge, Massachusetts Materials Systems Laboratory
©Jeremy Gregory and Randolph Kirchain, 2005 Materials Selection – Slide 15
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
= ≤
≥
=
⎡ ⎤= ⎢ ⎥
⎢ ⎥⎣ ⎦⎡ ⎤
= ⎢ ⎥⎣ ⎦
Massachusetts Institute of TechnologyCambridge, Massachusetts Materials Systems Laboratory
©Jeremy Gregory and Randolph Kirchain, 2005 Materials Selection – Slide 16
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
≥ = =
⎛ ⎞= ⎜ ⎟⎜ ⎟
⎝ ⎠
Massachusetts Institute of TechnologyCambridge, Massachusetts Materials Systems Laboratory
©Jeremy Gregory and Randolph Kirchain, 2005 Materials Selection – Slide 17
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
** ****
Massachusetts Institute of TechnologyCambridge, Massachusetts Materials Systems Laboratory
©Jeremy Gregory and Randolph Kirchain, 2005 Materials Selection – Slide 18
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.__________
Massachusetts Institute of TechnologyCambridge, Massachusetts Materials Systems Laboratory
©Jeremy Gregory and Randolph Kirchain, 2005 Materials Selection – Slide 19
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.__________
Massachusetts Institute of TechnologyCambridge, Massachusetts Materials Systems Laboratory
©Jeremy Gregory and Randolph Kirchain, 2005 Materials Selection – Slide 20
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
**
****