mech4301 l9 conflicting constraints - Τμήμα … · ppt file · web view2009-12-04 ·...

MECH4301 2007 Lecture # 9 Conflicting

Constraints 1/23

Advanced Methods in Materials Selection

•Conflicting Constraints

•Lecture 9 & Tutorial 4

• Conflicting Objectives

•Lecture 10 & Tutorial 5


Constraints 2/23

Tutorial 4: E7.2 and E7.3

Due Oct 1

Lecture 9: Chapters 9

& 10


Constraints 3/23

Outline Lecture 9

•Conflicting constraints:• “most restrictive constraint wins”• Case study 1: Stiff / safe / light column• Case study 2: Safe (no yield - no fracture)/ light air tank for a truck


Constraints 4/23

Multiple Constraints and Objectives • Design with multiple constraints

• Design with multiple objectives

One Objective:the performance metric

Rank by performance

metric

One Constraint

Many Constraints O

ne

Con

stra

int

Man

y C

onst

rain

ts

Mul

tiple

Obj

ectiv

es:

seve

ral p

erfo

rman

ce

met

rics

Trade-off and value function

method

Rank by most restrictive

performance metric

Function

Combinationof

methods

One Objective:one performance

metric

Rank by performance

metric

One Constraint

Many Constraints O

ne

Con

stra

int

Man

y C

onst

rain

ts

Mul

tiple

Obj

ectiv

es:

seve

ral p

erfo

rman

ce

met

rics

Penalty function method


performance metric

Function

Combinationof

methods

Minimise mass

Carry force Fwithout yielding,

given length

Tie rod

Simplest case:Design with one objective, meeting a single constraint

Or several non-conflicting constraints, such as melting point, corrosion resistance, etc.


Constraints 5/23

One Objective:one performance metric

Rank by performance

metric

One Constraint

Conflicting Constraints O

ne

Con

stra

int

Con

flict

ingC

onst

rain

ts

Con

flict

ing

Obj

ectiv

es:

conf

lictin

g pe

rfor

man

ce m

etric

s

Penalty function method


performance metric

Function

Combinationof

methods

One notch up in complexity: 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

Minimise mass

No yield &Given

deflectionNo corrosionTmax > 100C.

The most restrictive constraint determines the performance metric (mass)

Tie rodLecture 10


Constraints 6/23

Q7.1. Materials for a stiff, light tie-rod Constraint # 1

Minimise mass m: m = A L (2)

Objective

• Length L is specified• Must not stretch more than

Constraints

• Material choice• Section area A

Free variables

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

Strong tie of length L and minimum mass

L

FF

Area A

Tie-rod Function

m = massA = areaL = length = densityE= elastic modulus = elastic deflection

ELFm

2

1Performance

metric m1

Eliminate A in (2) using (1):

Chose materials with largest M1 =

E


Constraints 7/23

Q7.1. Materials for a strong, light tie-rod Constraint # 2

Minimise mass m: m = A L (2)

Objective (Goal)

• Length L is specified• Must not fail under load F

Constraints

• Material choice• Section area A

Free variables

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

Strong tie of length L and minimum mass

L

FF

Area A

Tie-rod Function

m = massA = areaL = length = density = yield strengthy

y

FLm

2Performance metric m2

Eliminate A in (2) using (1):

Chose materials with largest M2 =

y


Constraints 8/23

Q7.1: Conflicting Constraints: Strong /Stiff / Light Tie Rod

Ran

k by

the

mor

e re

stric

tive

of th

e tw

o, m

eani

ng…

?

Eval

uate

com

petin

g co

nstr

aint

s an

d pe

rfor

man

ce m

etric

s:

Max

. def

lect

ion

Mus

t not

yie

ld

yσσ

%1.0 %;1EL

= d

efle

ctio

n

y =

yie

ld s

treng

th

E =

ela

stic

mod

ulus

Competingperformance metrics

y

FLm

2

ELFm

2

1

Requires stiffer material

Stiffness constraintStrength constraint


Constraints 9/23

density E TS m1 (E) m2 (TS)Mg m-3 GPa MPa grams grams

Epoxy/Aramid Fibre, 1.38 80 1400 34.5 9.9Glass/Epoxy U-D Composite 1.95 45 1100 86.7 17.7Low alloy steel 7.9 217 2445 72.8 32.3Epoxy Carbon Fiber (SMC) 1.7 150 345 22.7 49.3Al 6061-T6 2.73 74 320 73.8 85.3Titanium 4.45 117 1138 76.1 39.1

Analytical solution in three steps: Rank by the more restrictive of the constraints

y

FLm

2

ELFm

2

11. Calculate m1 and m2 for given L and F

3. Find the smallest of the larger ones

The most restrictive constraint requires a larger mass and thus becomes the controlling or active constraint.

2. Find the largest of every pair of m’s


Constraints 10/23

0 0.02 0.04 0.06length (m )

0

0.02

0.04

0.06

0.08

mas

s (k

g)

m E = m

= 2700 Mg/m 3 E = 69 GPaF = 10000 Nt = 300 MPaL = 20 mm; /L = 0.1% ; /L = 1%

m

m E/L = 0.1%

m E/L = 1%

Graphical version of the analytical solution (for Aluminium)

mass

length

ELFm

2

1

y

FLm

2

y constraint active (heavier)

E constraint active (heavier) (long rod stretches too much)

/L= 1%: Strength constraint always active

Solution for /L= 1%

Less demanding E constraint => thinner rod


Constraints 11/23

Graphical solution using indices and bubble charts.

More general/powerful.Allows for a visual while physically based selection.Involves all available materials.Incorporates geometrical constraints through

coupling factors.

Pros to the graphical analytical solution : it makes explicit the dependence on L and L/.Cons: it is specific to the material considered (requires a dedicated graph per material)


Constraints 12/23


Constraints 13/23

Graphical solution using Indices and Bubble charts

21

1122

21 MFL

MLFFL

ELFmm

y

21 MLM

M2 =

y

M1 =

E

y

FLm

2

ELFm

2

1

make m1 = m2

Solve for M1

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

slope = 1 y-intcpt = L/

factorcouplingL

This is what we

know


Constraints 14/23


Constraints 15/23

Materials for High-Performance Con-Rods


Constraints 16/23


Constraints 17/23


Constraints 18/23

Density / Fatigue strength at 10^7 cycles10 100 1000

Dens

ity /

( 31

.62

* Yo

ung's

mod

ulus

0.5

)

10

100

1000

Wrought magnesium alloys

Titanium alloys

Cast magnesium alloys

Age-hardening wrought Al-alloys

Cast iron, gray

High F/L2

LowF/L2


Constraints 19/23

Density / Fatigue strength at 10^7 cycles1 10 100 1000 10000

Dens

ity /

( 31

.62

* Yo

ung's

mod

ulus

0.5

)

10

100

1000

Beryllium, grade I-250, hot isostatically pressed

Beryllium-aluminum alloy, beralcast 310, cast

Titanium, alpha-beta alloy, Ti-6Al-4V, cast

Magnesium, AZ61, wrought

Aluminum, 6061, wrought, T6

Magnesium, EA55RS, wrought


Constraints 20/23


Constraints 21/23

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

Youn

g's

mod

ulus

/ D

ensi

ty *

1e6

100

1000

10000

100000

Select with box on line of y-intercept = 100

Polyester SMC (Low Density)Phenolic/E-Glass Fiber, Woven Fabric Composite, Quasi-isotropic Laminate

Beech (Fagus grandifolia) (l)

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

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

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

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

E 7.1 tie rod Graphical solution (/L = 1% L/ = 100) Use level 3, exclude ceramics

y

2M

EM1

Simultaneously Maximise M1 and M2

m1 = m2

m1 < m2

m2 < m1

Coupling line for L/ = 100


Constraints 22/23Yield strength (elastic limit) / Density *1e3

1 10 100 1000

Youn

g's

mod

ulus

/ D

ensi

ty *

1e6

100

1000

10000

100000

Balsa (l) (ld)

Spruce (Picea abies) (l)

SiC/SiC Fiber, 35-45Vf - Woven LaminateBeryllium, grade I-250, hot isostatically pressed


Epoxy/HS Carbon Fiber, UD Composite, 0° Lamina

SiC/SiC Fiber, 35-45Vf - Quasi-isotropic Laminate Beryllium,

selection includes strong, light and very stiff composites like CFRP. Metals are generally excluded but for Be.


E7.1 Tie Rod Graphical solution ( /L = 0.1% L/=1000) Use level 3, exclude ceramics

EM1

y

2M



Active Constrain

t? 3-D view


Constraints 23/23

z = m ax(log10(1/x),log10(1/y))3-D view of the interacting constraints

EM1

y

2M

m1 = m2

m1 > m2m2 > m1

m2m1

lighter

•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


Constraints 24/23

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

Youn

g's

mod

ulus

/ D

ensi

ty *

1e6

100

1000

10000

100000


Polyester SMC (Low Density)Phenolic/E-Glass Fiber, Woven Fabric Composite, Quasi-isotropic Laminate

Beech (Fagus grandifolia) (l)

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

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


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

Graphical solution (deflection = 1% L/=100)

y

2M

EM1


m1 = m2

m2 < m1

m1 < m2

lighter


Constraints 25/23

Case Study # 2: Quite Similar to E7.2, Air cylinder for a truckDesign goal: lighter, safe air cylinders for trucks

Compressed air tank


Constraints 26/23

Case study: Air cylinder for truck

Free variables

Function Pressure vessel

Objective Minimise mass Constraints Dimensions L, R, pressure p, given

Safety: must not fail by yielding Safety: must not fail by fast fracture

Must not corrode in water or oil Working temperature -50 to +1000C

Wall thickness, t; choice of material

t

L

2R

Den

sity

Y

ield

stre

ngth

y

Frac

ture

toug

hnes

s K

1c

Pre

ssur

e p

Conflicting constraints lead to competing

performance metrics


Constraints 27/23

Air cylinder for truck

t

L

2R

Den

sity

Y

ield

stre

ngth

y

Frac

ture

toug

hnes

s K

1c

Pre

ssur

e p

)tR4LtR2(m 2

Obj

ectiv

e: m

ass

Vol of material in cylinder wall

f

2 SpLR2mEliminate t

f

2 SpLR2m*m

Stre

ss in

cyl

inde

r wal

l

StRp f

2

Failure stress

Safety factor

Asp

ect r

atio

,

)LR21(LtR2

May be either y or f transpose

What is the free variable?


Constraints 28/23

Air cylinder : graphical solution using CES charts

CES

Sta

ge 1

; app

ly s

impl

e (n

on c

onfli

ctin

g) c

onst

rain

ts:

wor

king

tem

p up

to 1

000 C

, res

ist o

rgan

ic s

olve

nts

etc.

Ran

k by

the

mor

e re

stric

tive

of th

e tw

o

y

m

*1

aKm

c /*

1

2

CES

Sta

ge 2

: eva

luat

e co

nflic

ting

perf

orm

ance

met

rics:

Mus

t not

yie

ld:

Mus

t not

frac

ture

yf 1

aK c

f 1

2

S =

safe

ty fa

ctor

a =

crac

k le

ngth

y =

yie

ld s

tren

gth

K1c

= F

ract

ure

toug

hnes

s

Competingperformance metrics for minimum mass


Constraints 29/23

Air cylinder - Simple (non- conflicting) constraints

CES Stage 1: •Impose constraints on corrosion in organic solvents•Impose constraint on maximum working temperature

Max service temp= 373 K (1000C)

Organic SolventsGood Very Good

Max

imum

Ser

vice

Tem

pera

ture

(K)

1000

Air cylinder

Low Carbon Steel

Wrought Al 1080-0

Wrought Al 2014, T4

Malleable cast iron

LA steel, AISI 4140 (normalised)

Epoxy - Glass Fibre

Epoxy - carbon

Corrosion resistance in organic solvents

K service .max T

Corrosion resistance

Select above this line


Constraints 30/23

Density / Yield strength (elastic limit) / 1e30.01 0.1 1

Dens

ity/

1000

/Fra

ctur

e to

ughn

ess

*((3

.14*

0.00

1)^

.5)

0.01

0.1

1

10

Titanium alloys Non age-hardening wrought Al-alloys

Copper

Wrought magnesium alloys

CFRP, epoxy matrix (isotropic)

Low alloy steelMedium carbon steel

Age-hardening wrought Al-alloysPolyamides (Nylons, PA) 2

1 and 1

1plot 21 Mm

Mm * *

Air cylinder - Conflicting constraints

y

m*

1

aK IC

m

*2

*m*m 21

*m*m 12 *2

* 1 mm

CES Stage 2: Find most restrictive constraint using Material Indices chart

Results so far:• 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

ICK

mE *2 :2.7


Constraints 31/23

Summary

• Real designs are over-constrained and many have multiple objectives

• Method of maximum restrictiveness copes with conflicting multiple constraints

• Analytical method useful but depends on the particular conditions set and lacks the visual power of the graphical method

End of Lecture 9

• Graphical method produces a more general solution

• Next lecture will solve air cylinder problem again for two conflicting objectives: e.g., weight and cost.


Constraints 32/23

There is a typographical error in textbook, Exercise E7.2, p. 581, 8 lines from the bottom

a / IcK

a IcKIt reads:

It should read:

mech4301 l9 conflicting constraints - Τμήμα … · ppt file · web view2009-12-04 ·...

Documents