2011 aula 5 selemat - PMT2051 aula 5 selemat - PMT2051.pdf · Title: Microsoft PowerPoint - 2011...

PMT 2501 – Análise de Falhas e Seleção de MateriaisCopyright : Granta, Ashby e Caceres

Aula 5- Restriçoes e objetivos múltiplos

One Objective:one performance metric

Conflicting Objectives:conflicting performance metrics

Function

one performance metric

Rank by performance

metric

One Constraint

Conflicting Constraints

One Constraint

ConflictingConstraints

conflicting performance metrics

Penalty functionmethod

Rank by most restrictive

performance metric

Combinationof

methods




Function

Single objective / Conflicting Constraints

Most designs are over-constrained: “Should not deflect more than something,

must not fail by yielding, by fatigue, by fast-fracture …” more constraints than

free variables

2/24


Rank by performance

metric

One Constraint


One Constraint





performance metric

Combinationof

methods

The most restrictive constraint determines the performance metric (mass)


Materials for a stiff, light tie-rod Constraint # 1

• Length L is specified

• Must not stretch more than δδδδConstraints

Equation for constraint on A: δ = Lσσσσ/E = LF/AE (1)

Strong tie of length L and minimum mass

L

FF

Area A

Tie-rodFunction

m = massA = areaL = lengthρ = density

Minimise mass m:

m = A L ρρρρ (2)Objective

• Material choice

• Section area A Free variables

δ = Lσσσσ/E = LF/AE (1) E= elastic modulusδ = elastic deflection

=

EL

Fm

ρ

δ2

1Performance

metric m1

Eliminate A in (2) using (1):

Chose materials with largest M1 =

ρ

E


Materials for a strong, light tie-rod Constraint # 2

• Length L is specified

• Must not fail under load FConstraints

Equation for constraint on A: F/A < σσσσ (1)

Strong tie of length L and minimum mass

L

FF

Area A

Tie-rodFunction

m = massA = areaL = lengthρ = density

Minimise mass m:

m = A L ρρρρ (2)

Objective (Goal)

• Material choice

• Section area A Free variables

F/A < σσσσy (1) ρ = density= yield strength

yσ

=

y

FLmσ

ρ2

Performance metric m2

Eliminate A in (2) using (1):

Chose materials with largest M2 =

ρ

σ y


Evaluate competing constraints and performance metrics:

Max. deflection

Must not yieldyσσ =

%1≤=EL

σσσσδδδδδ= deflection

σy = yield strength

E = elastic modulus

Materials for a stiff, light tie-rod Constraints # 1 and # 2

Competingperformance metrics

=

y

FLmσ

ρ2

=

EL

Fm

ρ

δ2

1

Stiffness

constraint

Strength

constraint


Graphical solution using Indices and Bubble charts

11FF ρρ

M2 =

ρ

σ y

M1 =

ρ

E

=

y

FLmσ

ρ2

=

EL

Fm

ρ

δ2

1

make m1 =m2

=

=

=

==

2

1

1

122

21M

FLM

LF

FLE

LF

mmy δσ

ρρ

δ

21 ML

Mδ

=Solve for M1

+=

δ

LMLogM log)2()1log( Straight line, slope = 1

y-intcpt = L/δδδδ

factorcouplingL

=δ


Young's m

odulus / Density *1e6

10000

100000

Beech (Fagus grandifolia) (l)

SiC/SiC Fiber, 35-45Vf - Woven Laminate

Cyanate Ester/HM Carbon Fiber, UD Composite, 0° Lamina

Phenolic/E-Glass Fiber, Woven Fabric Composite, Quasi-isotropic Laminate

Tie Rod Graphical solution (δδδδ/L = 1% => L/δδδδ = 100), level 3, excluding ceramics

=

ρ

EM

1

Simultaneously

maximise M1 and M2 m1 = m2

m1 < m2

Yield strength (elastic limit) / Density *1e31 10 100 1000

Young's m

odulus / Density *1e6

100

1000

Select with box on line of y-intercept = 100

Polyester SMC (Low Density)


selection includes several fibre reinforced composites more complaint than in the other case. Timbers are included.


=

ρ

σ y

2M

m2 < m1

Coupling

line for

L/δδδδ = 100

+=

δ

LMLogM log)2()1log(


z = max(log10(1/x),log10(1/y))

3-D view of the interacting constraints

=

ρ

EM

1

=σ yM

m1 = m2

m1 > m2 m2 > m1m2

m1

=ρ

M1

=

ρ

σ y

2M

lighterlighter

•m1 = m2 on the coupling line.

•The closer to the bottom corner, the lighter the component.

•Away from the coupling line, one of the constraints is active (larger m)

Locus of coupling

line depends on

coupling factor


Design goal: lighter, safe air cylinders for trucksCase study: Air cylinder for truck

Compressed air tank


Case study: Air cylinder for truck

Function Pressure vessel

Objective Minimise mass

t

L

2R

Density ρYield strength σy

Fracture toughness K1c

Pressure p

Free variables

Objective Minimise mass

Constraints Dimensions L, R, pressure p, given

Safety: must not fail by yielding

Safety: must not fail by fast fractureMust not corrode in water or oil

Working temperature -50 to +1000C

Wall thickness, t; choice of material

Conflicting constraints lead to competing

performance metrics


t

L

2R

Density ρYield strength σy

Fracture toughness K1c

Pressure p

ρπ+π= )tR4LtR2(m 2Objective: mass

Vol of material in cylinder wall Aspect ratio, αααα

)R2

1(LtR2 +ρπ=

What is the

free variable?


ρπ+π= )tR4LtR2(m 2Objective: mass

σ

ραπ=

f

2 SpLR2mEliminate t

=

f

mσσσσ

ρρρρ*

Stress in cylinder wallSt

Rp fσσ <=

2

Failure stress

Safety factor

)L

1(LtR2 +ρπ=

May be either

σσσσy or σσσσf !


CES Stage 1; apply simple (non conflicting) constraints:working temp up to 1000C, resist organic solvents etc.

CES Stage 2: evaluate conflicting performance metrics:

Must not yield: yσσ =f1

K

S = safety factor

a = crack length

σσσσy = yield strength


y

mσ

ρ=*

1

aKm

c /

*1

2

π

ρ=

Must not fracture

a

K c

1f2

πσ =

σσσσy = yield strength

K1c = Fracture toughness

Competingperformance metrics for minimum mass

=ρ

IcKM

2

=

ρ

σy

M1

Equate m1 to m2, and find the coupling factor for given crack size a.


CES Stage 1:•Impose constraints oncorrosion in organicsolvents•Impose constraint onmaximum workingtemperature

Ma

xim

um

Se

rvic

e T

em

pe

ratu

re (

K)

Air cylinder

Malleable cast iron

LA steel, AISI 4140 (normalised)

( )K service .max T

Select above

this line


Max service temp= 373 K (1000C)

Organic Solvents

Good Very Good

Ma

xim

um

Se

rvic

e T

em

pe

ratu

re (

K)

1000

Low Carbon Steel

Wrought Al 1080-0

Wrought Al 2014, T4

Malleable cast iron

Epoxy - Glass Fibre

Epoxy - carbon

Corrosion resistance in organic solvents


CES, Stage 2: Equate m1 to m2, and find the coupling factor

for given crack size a.

aaK Ic πσ

ρ

π

ρ 1 factor coupling

1 ==>=

y

mσ

ρ=*1

aKm

c /

*1

2

π

ρ==

=ρ

IcKM

2

=

ρ

σy

M1

)log()1

log()log( 12

11

−− += Ma

Mπ

for a crack a = 5 mm, the coupling factor is 1/√(3.14*0.005) = 1/0.12 = 8


Density/ 1000/Fracture toughness

1

10

100

coupling line at M= 8, for 5 mm crack

Epoxy/S-Glass Fiber, UD Composite, 0° Lamina

Cyanate Ester/HM Carbon Fiber, UD Composite, 0° Lamina

fibre reinforced composites are the best materials, steels and many alloys are selected too

Epoxy/E-Glass Fiber, Woven Fabric Composite, Biaxial Lamina

c = 5 mm crack*m*m 12 >

*

2

*

1 mm =

Results so far:• Epoxy/carbon fibre

=

−

IcK

Mρ1

2

a = 5 mm intcpt= 8 @ 1/M1= 1

Density /Yield strength (elastic limit) /1e31e-3 0.01 0.1 1

Density/ 1000/Fracture toughness

0.01

0.1

Phenolic/E-Glass Fiber, Woven Fabric Composite, Biaxial Lamina

Wrought austenitic stainless steel, AISI 302, HT grade B

7075, T761 Aluminum/Aramid Fiber, UD Composite, 0° Lamina

2

1 and

1

1plot

21

Mm

Mm

* *

∝∝

=

−

y

Mσ

ρ1

1

*m*m 21 >

• Epoxy/carbon fibre composites• Epoxy/glass fibre composites• Low alloy steels• Titanium alloys• Wrought aluminium alloy• Wrought austenitic stainless steels• Wrought precipitation hardened stainless steels

Lighter this way




Function

Conflicting Objectives and Single constraint

Most designs also present conflicting objectives: cost, weight, CO2

emission, volume...


Rank by performance

metric

One Constraint


One Constraint





performance metric

Combinationof

methods


Examples of Conflicting Objectives in design

� Common design objectives:

Minimising mass (sprint bike; satellite components)

Minimising volume (mobile phone; minidisk player)

Minimising environmental impact (packaging, cars)

Minimising cost (everything)

Objectives

Some objectives may mass, m conflict with another cost, c

We wish to minimize both (all constraints being met)

� Conflict : the choice that optimises one does not optimise the other.

� Best choice is a compromise (strategies).

Each objective defines a performance metric


Me

tric

2:

Co

st

CE

xpensiv

e

Multi-objective optimisation: The terminology• Solution: a viable choice,meeting constraints, but notnecessarily optimum by eithercriterion.

• Plot all viable solutions as function of performance metrics. (Convention: expressobjectives to be minimised)

A Dominatedsolution

B Non-dominatedsolution

Light Metric 1: Mass m Heavy

Cheap

Me

tric

2:

• Trade-off surface: the surface on which the non-dominated solutions lie (also called the Pareto Front) (after Pareto, 1898)

Trade-offsurface

• Dominated solution: one thatis unambiguously non-optimal(as A) (there are better ones)

• Non-dominated solution: onethat is optimal by one metric (asB: optimal by one criterion butnot necessarily by both)


Example of Conflicting Objectives in Pushbikes

Price vs. mass of bicycles: a matter of perception?

Price $

Mass (kg)


Strategy 1: compromise by intuition and experience

• Make trade-off plot and Sketch trade-off surface

• Use intuition to select a solution on the trade-off surface

Metr

ic 2

: C

ost

C

Exp

en

siv

e

Trade-off

select

• “Solutions” on or near the surfaceoffer the best compromisebetween mass and cost

•The choice depends on how highly you value a light weight, a question ofrelative values


Ch

ea

pM

etr

ic 2

:

Trade-offsurface

current material


Strategy 2, finding a compromise:

• Reformulate all but one ofthe objectives as constraints,setting an upper limit for it

Me

tric

2:

Co

st

CE

xpensiv

e

Trade-offsurface

Mass and price of bicycles:

Upper limit for cost: $200.

Optimum solutionminimising m


Cheap

Me

tric

2:

• Good if you have budget limit

• Trade-off surface leads you to thebest choice within budget

• But not a true optimisation --mass has been treated as aconstraint, not an objective.

Optimum solutionminimising c

Constraint: mass = 11 kg


Me

tric

2:

Co

st

CE

xpensiv

e

Strategy 3: Penalty functions and exchange constants

Z1

Z2Z3

Z4 Contours of constant Z

Decreasingvalues of Z

Seek material with smallest Z($). Cost of decrease weight,decrease CO2 foot-print.....

Define locally linearPenalty function Z ($)

Cm += αZ


Cheap

Me

tric

2:

Optimum solution,minimising Z

(lowers both m and c)

values of Z

α−

decrease CO2 foot-print.....

Make a trade-off plot

But what is the meaning of αααα ?

• plot on it contours of Z

- lines of constant Z haveslope -α

ZmC +−= α

• Read off solution with lowest Z


Co

st

CE

xpensiv

e

Z = penalty function.

Z1

Along the line

Z = cost + αααα mass = constant


Cheap

Me

tric

2:

Co

st

C

α−

cost

mass

Z is the combined “value” of cost & mass


The exchange constant αααα

The quantity αααα is called an “exchange constant”-- it measures the value of performance, here theprice of saving 1 kg of mass ($/kg).

Cm

∂∂

=ZαCmZ += α

αααα = drop in Z

per unit mass, at

constant cost

Me

tric

P2

: C

os

t C

price of saving 1 kg of mass ($/kg).

How get a value for α…?

� market survey (perceived value)

� full life cost (engineering criteria)

� guess…..

Metric P1: Mass m


Family car (based on fuel saving)

Truck (based on payload)

Civil aircraft (based on payload)

Transport System: mass saving αααα ($US per kg)

0.5 ~ 6

5 to 20

100 to 500

Exchange constants for mass saving in transport systems

Example of values of the exchange constant (αααα) for transport systems

Savings over 2x105km

Civil aircraft (based on payload)

Military aircraft (performance payload)

Bicycle frame (perceived value)

Space vehicle (based on payload)

100 to 500

500 to 1000

20-4000

3000 to 10000

The is how much you can afford to expend in a material substitution in terms of weight. If the substitution costs you more than the upper bound, you won’t get your money back.

C. H. Cáceres, "Economical and environmental factors in light alloys automotive applications", Metall. Mater. Trans. A, 2007, 38, 1649-1662. M. F. Ashby, "Multy-objective optimization in material design and selection", Acta Materialia, 2000, 48, 359-369.


Penalty function on log scales

Log scales

Exp

en

siv

e

� A linear relation, on log scales,plots as a curve ZmαC

CmαZ+−=

+=

Linear scales

Exp

en

siv

e

Lighter mass, m Heavier

Ch

ea

p

C

ost,

C

Decreasing values of Z

Lighter mass, m Heavier

Ch

ea

p

C

ost,

C

Decreasing values of Z

-αααα


De

nsity x

Pri

ce

/S

qrt

Mo

du

lus

1000

10000

100000

1e6

GFRP

Epoxy/HS Carbon

weave Lead alloys

Copper alloys

Tungsten alloys

BronzeCFRP epoxy

laminate

Ti-alloys

Ni-based superalloys

Cobased superalloys

P2= Costfor givenstiffness

Exchange constant

α = 500 $/kg

Trade-off surface

ρρρρc/E1/2

Family car

Truck

Civil aircraft

Military aircraft

Bicycle frame

Space vehicle

System αααα ($US per kg)

0.5~6

5 to 20

100 to 500

500 to 1000

20-4000

3000 to 10000

27/30

Density/Sqrt Modulus50 100 200 500 1000 2000 5000

De

nsity x

Pri

ce

/S

qrt

Mo

du

lus

10

100

1000

MAGNESIUM alloys

ALUMINUM alloys

HSLA steels CAST IRONS

Zinc alloys

Copper alloys

Penalty function in transport systems.Mass of a beam vs. cost for given stiffness

P1=Massfor givenstiffness

Exchange constant

α = 1 $/kg

ρρρρ/E1/22P

∆

∆−=

1P2

Pα


Case study: casing for electronic equipment

� Electronic equipment -- portable computers,players, mobile phones, cameras – areminiaturised; many less than 12 mm thick

� Minidisk player: An ABS or Polycarbonatecasing has to be > 1mm thick to be stiff enoughto protect; casing takes 20% of the volume

stiff, light, thin casingFunction stiff, light, thin casing

bending stiffness EI at least that of existing case

minimise casing thicknessminimise casing mass

choice of materialcasing thickness, t

Constraints

Objectives

Function

Free variables

The thinnest may not be the lightest … need

to explore trade-off


Performance metrics for the casing: t and m

Function Stiff casing

t

w

L

F

m = massw = widthL = lengthρ = density

3L

IE48S =

Constraints

12

twI

3

=

� Adequate toughness, G1c > 1kJ/m2

� Stiffness, S

with

Metric 1 3/1

3/13

E

1

wE4

LSt ∝

=

Objective 2 Minimise mass m

Metric 23/13/1

2

3/12

EEL

C

wS12m

ρ∝

ρ

=

ρ = densityt = thicknessS = required stiffnessI = second moment of areaE = Youngs Modulus

Objective 1 Minimise thickness t

Unit 5, Frame 5.10

Materials Index to minimise the thickness

Materials Index to minimise the mass


Relative performance metrics

� The thickness of a casing made from an alternative material M, differs (for the same stiffness) from one made of Mo by the factor

3/1o

o E

E

t

t

=

We are interested here in substitution. Suppose the casing iscurrently made of a material Mo, elastic modulus Eo, density ρo.

Relative thickness = ratio of Materials Indices (t)

� The mass differsby the factor

ρ

ρ=

o

3/1o

3/1o

E.

Em

m

om

m� Explore the trade-off between and

ot

t

� Define a relativepenalty function, Z* oo t

tmm ** αZ += (α now dimensionless)

Relative mass = ratio of Materials Indices (mass)


Plotting the relative penalty function, Z*

� Penalty lines for casing

Assume mass and thickness are equally important: α* = 1

** Z+−=oo tt

mm α

10

Ma

ss

re

lati

ve

to

AB

S

Materials on trade-off surface are

metals and high performance composites

Penalty functions

of gradient -αααα* = -1

α* = ???

Current casing

31/30Thickness relative to ABS

0.1 1 10

Mass r

ela

tive

to A

BS

1

Low alloy steel

Al-alloys

Mg-alloys

GFRPCFRP

Al-SiC Composites

Ti-alloys

ABSNi-alloys

Thickness relative to ABS

Ma

ss

re

lati

ve

to

AB

S

Z*1Z*2

Z*3


Mass re

lative

to A

BS

1

10

PTFE

PC

ABS

PMMA

PP

PE

Ionomer Ni-alloys

Cu-alloys

Steels

Al-alloys

Ti-alloys

Lead

Elastomers

Mass r

ela

tive

to A

BS

, m

/mo

Trade-offsurface

Conclusion: Four-sector trade-off plot for minidisk player

� The four sectors of a trade-off plot for substitution

D. Worse by both metrics

B. Thinner but heavier

win-lose sector: worth exploring

Don’t bother

Current casing

32/30

Thickness relative to ABS0.1 1 10

Mass re

lative

to A

BS

0.1

PP

NylonPolyesterAl-SiC Composite

Mg-alloys

CFRP

GFRPPolymer foams

.

Thickness relative to ABS, t/to

Mass r

ela

tive

to A

BS

, m

/m

Q: Is material cost relevant? Not a lot -- the case only weighs a few grams. Volume and weight are much more valuable.

A. Better by both metrics

C. Lighter but thicker

win-win sector

win-lose sectors: worth exploring


The main points

� Real design problems involve conflicting objectives -- often technical or

environmental performance vs. economic performance (cost).

� Trade-off plots reveal the options for material selection or materialsubstitutions that solve the conflict, and (when combined with the otherconstraints of the design) frequently point to a sensible final choice.

� If the relative value of the two metrics of performance (measured by anexchange constant) is known, a penalty function allows an unambiguousselection: the exchange constants allow exploring the chart's win-lose(trade-off) sectors as well as the win-win sector.

2P

∆

∆−=

1P2

Pα

Engineering

definition of αααα

P1, P2 = performance

metrics (mass, cost,

volume, CO2)

2011 aula 5 selemat - PMT2051 aula 5 selemat - PMT2051.pdf · Title: Microsoft PowerPoint - 2011...

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