ifb 2012 introduction material indices1/12 ifb 2012 materials selection in mechanical design...

IFB 2012 INTRODUCTION Material Indices 1/12

IFB 2012Materials Selection in Mechanical Design

INTRODUCTION

Materials Selection Without Shape (1/2)

Textbook Chapters 5 & 6

Shape of cross section is kept

constant. Only the material changes.

b2 = Free variable

(Trade-off variable)


Deriving Materials Indices, Example 1: Material for a stiff, light beam

2/1

2/15

ECLS12

m Chose materials with largest M=

2/1E

Material Index• Material choice

• Area = b2 What can be

varied ? {

I = second moment of area:12b

I4

Beam (solid square section)

Function

m = mass A = cross section

L = length = density

b = edge length S = stiffness I = second moment of area E = Young’s modulus

Stiffness of the beam, S:

3L

IECS

δ

FS

Constraint

LbLAm 2Minimise mass, m:Goal

b2 = m/L b4 = 12 SL3/CE

To minimise the mass Maximise Material Index !

Get these equations from the textbook,

or pdf file “Useful

Solutions”


Elastic Bending of Beams and Panels; p. 533 or from pdf file “useful solutions”

31

L

EICFS

3S

L

IEC

??I

δ

FS


Moments of Sections; p 531, or file “Useful Solutions”

12b

I4


m = massw = widthL = length = densityt = thicknessS = stiffnessI = second moment of areaE = Young’s modulus

Panel, width w and length L specified

Stiffness of the panel, S

3L

IECS

12tw

I3

3/12

3/12

EL

CwS12

m

LtwLAm

Minimise mass, m

Chose materials with largest M=

3/1E

δ

FS

Deriving Materials Indices, Example 2: Material for a stiff, light panel

Material Index

Function

Goal

Constraint

• Material choice.• Panel thickness t

What can be varied ? { Eliminate t

Free variable..

??


Material Indices for Minimum Mass

Function Index

Same Volume

Tension (tie)

Bending (beam)

Bending (panel)

E/ρ

/ρE1/2

ρE1/3 /

1/ρ

Objective: minimisemass for given stiffness

Objective: minimise mass


MECH4301 2011 Lecture 2 Charts 7/12

Materials Selection using charts: effect of slope of selection line

Index ME

M

E

2/1

ME

3/1

Selection line slope

2


1


3

Different materials are selected,

depending on the slope of the selection

line

Exam question: What is the

Physics behind the different

exponents in the Indices’

equations?


Demystifying Material Indices

This is how the world looks like

after you pass the Materials

Selection Course


Demystifying Material Indices (beam, elastic bending)

2/11

1

2/15

1

12

EC

LSm

2/12

2

2/15

2

12

EC

LSm

2

1

1

2/11

2/12

2

1

2

M

ME

Em

m

For given shape, the reduction in mass at constant bending stiffness

is given by the reciprocal of the ratio of material indices.

Same applies to bending strength.

Material 1, Mass 1 stiffness S

Materials 2, Mass 2 stiffness S


Example: How good are Mg and Al when it comes to reducing mass?

E (GPa)

(Mg/m3) Tie-rod Beam Panel

Equal Volume

Steel 210 7.8 10 10 10 10

Al 75 2.7 10 5.9 4.9 3.5

Mg 44 1.7 11 5.1 3.9 2.2

3/1EE

A 10 kg component made of Steel…

heavier lighter

2/1ESteel

Mg

m2/m1 = M1/M2

Exam question: Which beam is fatter?? Same: Which panel is thicker??

Comparative weight of panels of equal stiffness (Steel, Ti, Al and Mg) (Emley, Principles of Mg Technology)


E (GPa)

(Mg/m3)

Relative weight

MgLi 44 1 3

Mg 44 1.7 4

Al 75 2.7 5

Ti 115 4.5 7

Steel 210 7.8 10

E (GPa)

(Mg/m3)

Relative mass

MgLi 44 1 3

Mg 44 1.7 4

Al 75 2.7 5

Ti 115 4.5 7

Steel 210 7.8 10

The Mg-Li panel is thicker


Example of solution to Tutorial # 1 (Exercise 7.3)

Derivation of the Material Index:When fully loaded, the beam should not fail, i.e., maximum < * (yield strength)

m = lA Solve for A= m/l.

The maximum force isI/ym =A3/2/6

2011 Lecture 3 Material Indices 13/12

2/3

*

2/5

2/3*

6

l

mC

ly

ICF

m

3/2*

3/53/23/2

)(

6

LFC

mSolving for m:

Select using the - chart with a line of slope 1.5, on the upper left corner.

Material Index : M = (*)2/3/.

Example of solution to Tutorial # 1 (Exercise 7.3)

2011 Lecture 3 Material Indices 14/12

Name X-Axis Y-Axis Stage 1: IndexCFRP, epoxy matrix 1500 - 1600 550 - 1050 0.05375Wood, typical along grain 600 - 800 60 - 100 0.02626Flexible Polymer Foam 16 - 35 0.24 - 0.85 0.02488Magnesium alloys 1740 - 1950 185 - 475 0.02414Polyamides (Nylons, PA) 1120 - 1140 90 - 165 0.02175

Rigid Polymer Foam (LD) 36 - 70 0.45 - 2.25 0.02Silicon carbide 3100 - 3210 400 - 610 0.01981GFRP, epoxy matrix 1750 - 1970 138 - 241 0.01732Titanium alloys 4400 - 4800 300 - 1625 0.01713Bamboo 600 - 800 36 - 45 0.01695Flexible Polymer Foam (LD) 38 - 70 0.24 - 2.35 0.01602

Alumina 3800 - 3980 350 - 588 0.01518Rigid Polymer Foam (MD) 78 - 165 0.65 - 5.1 0.01314Stainless steel 7600 - 8100 480 - 2240 0.01306Polyester 1040 - 1400 41.4 - 89.6 0.01283Low alloy steel 7800 - 7900 550 - 1760 0.0126High carbon steel 7800 - 7900 550 - 1640 0.01261Flexible Polymer Foam 70 - 115 0.43 - 2.95 0.01207Aluminum alloys 2500 - 2900 58 - 550 0.01178

Density (kg/ m^3)100 1000 10000

Tensi

le s

trength

(M

Pa)

1

10

100

1000

Wood, typical along grain

Magnesium alloys

Silicon carbide

CFRP, epoxy matrix (isotropic)

Bamboo

Polyamides (Nylons, PA)

Magnesium alloys

Rigid Polymer Foam (HD) Aluminum alloys

Titanium alloys

Rigid Polymer Foam (MD)

Flexible Polymer Foam (LD)

Stainless steel

GFRP, epoxy matrix (isotropic)

Selection line gradient1.5

Copy the results from CES

Conclusions to the chart/table: Composites, timber are the best materials. Al, Mg and steels are good competitors. Foams perform generally well, due to their low density. However, if made out of foams, the beams will be rather fat/big!

Select using the - chart with a line of slope 1.5, upper left corner. Sort the materials by their Index


The End Introduction


The CES software: Demonstration


Organising information: the MATERIALS TREE

Kingdom

Materials

Family

• Ceramics& glasses

• Metals & alloys

• Polymers & elastomers

• Hybrids

Class

Steels

Cu-alloys

Al-alloys

Ti-alloys

Ni-alloys

Zn-alloys

Member

10002000300040005000600070008000

A material record

Attributes

Density

Mechanical props.

Thermal props.

Electrical props.

Optical props.

Corrosion props.

Supporting information

-- specific

-- general

Density

Mechanical props.

Thermal props.

Electrical props.

Optical props.

Corrosion props.

Supporting information

-- specific

-- general


CES : the 3 levels

Level 2 enough for most exercises

3400


Chart created with the CES software (level 1, 60 materials)

Density (kg/m^3)100 1000 10000

Youn

g's

mod

ulus

(G

Pa)

1e-3

0.01

0.1

1

10

100

Rigid Polymer Foam (HD)

Flexible Polymer Foam (VLD)

Silicone elastomers

Polyvinylchloride (tpPVC)

Wood, typical along grain

Tungsten carbidesAlumina

Silicon

Zinc alloys

Lead alloys

Tungsten alloys

Concrete

Brick

CFRP, epoxy matrix (isotropic)

Rigid Polymer Foam (MD)

Rigid Polymer Foam (LD)

Cork

Polychloroprene (Neoprene, CR)

Leather

Wood, typical across grain

E

density


Chart created with the CES software (level 3, ~3400 materials)

Density (kg/m^3)10 100 1000 10000

Youn

g's

mod

ulus

(G

Pa)

1e-5

1e-4

1e-3

0.01

0.1

1

10

100

1000

Melamine Foam

Ethylene-Propylene Rubber (EPM)

Silicone elastomer (Shore A40)

Diamond

Balsa (l) (ld)

Leather

Tin-Lead 63-37 Solder

Metal Impregnated Carbon

Silicone elastomer

Polyurethane Foam

Polymethacrylimide Foam:

Palm (0.35)

E

density


Ranking Materials using Charts

CE

2/1

CE

CE

3/1

1

2

3

E

metals

ceramics

composites & polymers

foams

Selection line for tie rodsSelection line

for beams

Selection line for panels

One very significant conclusion from this course, so far: For beams

and panels, materials with very low density are more important

than for tie-rods. This is why foams are not used for tie rods, but are preferred for beams and

more so for flat panels.

Selection corner

Important: Read textbook pp.93-95: Summary and Conclusions to Ch. 4,

Properties of charts.


mass = xVprice [c ] = $/kgTotal cost C = c x mass = c VTotal cost C c [$/m3]

Function Index

Tension (tie)

Bending (beam)

Bending (panel)

cρE/

cρ/E1/2

ρE1/3 c/

Material Indices for Minimum Cost?

Goal: minimise cost

Performance metric = cost per given stiffness

To minimise the cost

Maximise Material Index !

Same for embodied energy Q = q, etc.

[ c ] = [$/m3] “price density”


Comparative stiffness of panels of equal weight (Steel, Ti, Al and Mg) (Emley, Principles of Mg Technology)

E (GPa)

(Mg/m3)

Relative stiffness

MgLi 44 1 23

Mg 44 1.7 19

Al 75 2.7 8

Ti 115 4.5 3

Steel 210 7.8 1


Material for a stiff tie-rod of minimum mass

Minimise mass m: m = A L

• Length L is specified• Must not deflect more

than under load F

• Material• Cross section area A

Equation for constraint on : ≤ L = L /E = L F/A E

Chose materials with largest M =

E

Material Index

E

FLm

2

Constraints

Goal

What can be varied to meet the goal ?

Performance metric: mass

{

{

A = LF/E

m = massr = densityE = Young’s modulus = deflection

Tie-rodFunction

To minimise the mass

Maximise Material Index !

A = Free variable ; or

Trade-off variable


Materials for a strong, light beam

Lm

bA 2

m = massA = areaL = length = densityMf = bending strengthI = second moment of areaE = Youngs ModulusZ = section modulus

Beam (shaped section).

Bending strength of the beam Mf:

Combining the equations to eliminate A gives:

** Zy

IM

mf

3/2*

3/53/26

LFm f

LAm


3/2*

Minimise mass, m, where:

Function

Objective

Constraint

Area A

88

32 bbhZ



Failure of Beams; p. 535


Moments of Sections; p 531


Materials for a strong, light tie-rod

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

FLmPerformance metric m


y

Eliminate A in (2) using (1):



ExampleObjective: minimise mass

Performance metric = mass

Function Stiffness

Strength

Tension (tie)

Bending (beam)

Bending (panel)

E/ρ ρ/y

/ρE1/2 ρσ2/3 /y

ρE1/3 /ρσ1/2

y /

Material Indices

An objective defines a performance metric: e.g. mass or cost.

The equation for the performance metric contains material properties. Sometimes a single property

Sometimes a combinationEither is a material index


Material Indices

Each combination of

FunctionConstraintObjectiveFree variable

has a characterising material index

Maximise this!

INDEX

yM

INDEX

2/1EM

Maximise this!


The End Introduction

ifb 2012 introduction material indices1/12 ifb 2012 materials selection in mechanical design...

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