Design Notes3 (6)

7/28/2019 Design Notes3 (6)

1/34

The Design Core MarketAssessment

Speci f icat ion

Concept

Design

Detail

Design

Manufacture

Sell

DETAIL

DESIGN

A vast subject. We will concentrate on:

Materials Selection

Process Selection

Cost Breakdown


2/34

Materials Selection with Shape

FUNCTION

MATERIAL

PROCESS

SHAPE

SHAPES FOR TENSION,BENDING, TORSION,

BUCKLING

--------------------

SHAPE FACTORS

--------------------

PERFORMANCE INDICES

WITH SHAPE


3/34

Common Modes of Loading


4/34

Moments of Sections: Elastic

SectionShape

A(m2)

I(m4)

K(m4)

2r 4

4r

4

2r

2b

12

4b414.0 b

ab ba3

4

)( 22

33

ba

ba

bh12

3bh

)(

58.013

3

bh

h

bhb

2

4

3a

332

4a

80

34a

rt

rr io

2

)( 22

tr

rr io

3

44 )(4

tr

rr io

3

44

2

)(2

A = Cross-sectional area

I= Second moment of area

tion ytion ybyAyI sec2

sec

2 dd

where yis measured verticallyby is the section width at y

K= Resistance to twisting of section

( Polar moment Jof a circular section)

tion rrJ sec3d2

GTLK

where Tis the torque

L is the length of the shaft

is the angle of twist

G is the shear modulus


5/34

Moments of Sections: Elastic

bt4

tba )(

bt

hhb io

2

)(

)(2 bht

)(2 bht

2

22

41

dt

tb3

3

2

a

bta

31

4

3

2/

)(12

2

33

o

io

bth

hhb

h

bth

31

6

1 3

)4(6

23 btht

8

2dt

4

3 1

b

ttb

22

2/5)(4

ba

tab

bh

htb

222

b

hbt

41

3

2 3

)8(3

3

hbt

h

bht

41

3

2 3

SectionShape

A(m2)

I(m4)

K(m4)


6/34

Moments of Sections: Failure

SectionShape

Z(m3)

Q(m3)

3

4r

3

2r

6

3b 321.0 b

ba2

4

)(

2

2

ba

ba

6

2bh

)(

8.13

22

bh

bh

hb

32

3a

20

3a

tr

rrr

io

o

2

44 )(4

tr

rrr

io

o

2

44

2

)(2

Z= Section modulus

my

IZ

where

ym is the normal distance from the neutral axisto the outer surface of the beam carrying the

highest stress

Q = Factor in twisting similar to Z

TQ

where

is the maximum surface shear stress


7/34

Moments of Sections: Failure

SectionShape

Z(m3)

Q(m3)

tb2

3

42

2 12

b

ttb

a

bta 31

4

2

)(

)(2 2/13

ab

bat

o

io

o

bth

hhh

b

336

h

bth 31

3

2

tbh2

b

hbt

41

3

2 2

)4(3

23 bthh

t )8(

3

2

hbt

h

bht

41

3

2 2

4

dt


8/34

Shape Factors: Elastic

BENDING

3

1

L

EICSB

44

24 ArIo

Bending stiffness of a beam

where C1 is a constant depending on the

loading details, L is the length of the beam,

and Eis the Youngs modulus of the material

oBo

Be

B I

I

S

S

Define structure factor as the ratio of the

stiffness of the shaped beam to that of a

solid circular section with the same cross-

sectional area thus:

2

4

A

IeB

so,

TORSION

L

KGST Torsional stiffness of a beam

where L is the length of the shaft, G is the

shear Modulus of the material.

22

24 ArKo 2

2

A

KeT

so,

oTo

Te

T K

K

S

S

Define structure factor as the ratio of the

torsional stiffness of the shaped shaft to that

of a solid circular section with the same

cross-sectional area thus:


9/34

Shape Factors: Failure/Strength

BENDING

44

2/33 ArZo

ofo

ff

BZZ

MM

Define structure factor as the ratio of the failure

moment of the shaped beam to that of a solid

circular section with the same cross-sectional area

thus:

2/3

4

A

ZfB

so,

Z

M

I

Mym

ff ZM

The beam fails when the bending moment is largeenough forto reach the failure stress of the

material:

The highest stress, for a given bending moment M,

experienced by a beam is at the surface a

distance ym furthest from the neutral axis:

TORSION

Q

T

The highest shear stress, for a given torque T,

experienced by a shaft is given by:

22

2/33 ArQo 2/3

2

A

QfT

so,

ofo

ff

TQQ

TT

Define structure factor as the ratio of the failure

torque of the shaped shaft to that of a solid circular

section with the same cross-sectional area thus:

ff QT

The beam fails when the torque is large enough forto reach the failure shear stress of the material:


10/34

Shape Factors: Failure/Strength

Please Note:

The shape factors for failure/strength described in this lecture course are

those defined in the 2nd Edition of Materials Selection In Mechanical

Design by M.F. Ashby. These shape factors differ from those defined inthe 1st Edition of the book. The new failure/strength shape factor

definitions are the square root of the old ones.

The shape factors for the elastic case are not altered in the 2nd Edition.


11/34

Comparison of Size and Shape

Rectangular sections

I-sections

SIZE


12/34

Shape Factors

SectionShape

Stiffness Failure/StrengtheB

eT fB

f

T

11 11

05.1

3

0.88 0.7418.1

3

2

b

a

b

a)( ba

b

a

22

2

ba

ab

b

h

3

)(

58.013

2

bh

b

h

h

b

2/1

3

2

b

h

)(

)/6.01(3

)/(22

2/1

bh

hb

hb

0.6221.133

2

73.0

35

2

2/12

t

r

t

r

t

r2/1

2

t

r

0.77


13/34

Shape Factors contd

SectionShape

Stiffness Failure/StrengtheB

eT fB

f

T

t

b

6

4

18

b

t

t

b

2

)/1(

)/31(

abt

aba

222

2/5

))((

)(8

babat

ab

bt

h

2

2

t

d

2

2

2)/1(6

)/31(

hbt

hbh

2

32

)/1(6

)/41(

hbt

hbth

3

22

)( bht

hb

2)/1(3

)/41(

bhb

bht

2)/1(6

)/81(

hbh

hbt

2)/1(3

)/41(

hbh

hbt

2/1

3

2

t

b22/1

12

b

t

t

b

2/3

2/1

)/1(

)/31(

ab

ab

t

a

2/32/1

2/1

)/1(

4

bat

a

2/1)(

2

bt

h

2/3

2/1

)/1(

)/31(

3

2

hb

hb

t

h

2/3

322/1

)/1(

)/41(

2 hb

hbt

t

h

2/1)(

t

d

2/32/1 )/1()(

2

bhbt

h

2/3

2/1

)/1(

)/41(

3

2

hb

bh

b

t

2/3

2/1

)/1(

)/81(

18 hb

hb

h

t

2/3

2/1

)/1(

)/41(

3

2

hb

hb

h

t


14/34

Efficiency of Standard Sections

2

4

A

IeB

ELASTIC BENDING

Shape Factor:

4loglog2log

e

BAI

Rearrange forIand take logs:

Plot logI against logA: parallel lines of slope 2eB


15/34


2/3

4

A

ZfB

BENDING STRENGTH

Shape Factor:

4loglog

2

3log

f

BAZ

Rearrange forIand take logs:

Plot logI against logA: parallel lines of slope 3/2fB


16/34


ELASTIC TORSION

2loglog2log;

22

e

Te

T AKA

K

TORSIONAL STRENGTH

2loglog

2

3log;

22/3

f

Tf

T AQA

Q

N.B. Open sections are good in bending, but poor in torsion


17/34

Performance Indices with Shape

ELASTIC BENDING

3

1

L

EICSB Bending stiffness of a beam:

ELASTICTORSION

L

KGST Torsional stiffness of a shaft:

2/12

2/5

2/1

1

4

e

B

B

EL

C

Sm

f1(F) f2(G) f3(M)

2/1

1

)( eBEM So, to minimize

mass m, maximise

2

4

A

IeB

Shape factor:

3

2

1

4 L

AECS

e

BB

so,

2/1

2

)( eTEM

So, to minimize

mass m, maximise

2

2

A

KeT

Shape factor: L

AG

S

e

T

T

2

2

so,

2/1

22/32/12

e

T

TG

LSm

f1(F) f2(G) f3(M)

EG8

3


18/34

Performance Indices with Shape

FAILURE IN BENDING

ff ZM Failure when moment reaches:

FAILURE IN TORSION

ff QT Failure when torque reaches:

3/2

3

)( fBfM So, to minimize

mass m, maximise

2/3

4

A

ZfB

Shape factor:

4

2/3AM

f

Bff so,

3/22/3

3/2)4(

f

Bf

f LMm

f1(F) f2(G) f3(M)

3/2

4

)( fTfM So, to minimize

mass m, maximise

3/2

2/33/2

4

f

Tf

f LTm

f1(F) f2(G) f3(M)

2/3

2

A

QfT

Shape factor:

4

2/3AT

f

Tff so,

)2( ff


19/34

Shape in Materials Selection Maps

0.01

0.1

1

10

100

1000

0.1 1 10 100

Density, (Mg/m3)

Y

oung'sModulus,E

(GPa) Engineering

Alloys

Polymer

Foams

Woods

Engineering

Polymers

Elastomers

Composites

CeramicsSearch

Region

A material with Youngs modulus,Eand density,, with a particular

section acts as a material with aneffective Youngs modulus

and density

e

BEE

e

B

Performance index for elastic

bending including shape,

can be written as

2/1

1

)( EM

e

B

e

B

e

BEM

2/1

1

)(

EXAMPLE 1, Elastic bending

=1

=10


20/34

Shape in Materials Selection Maps

A material with strength, fand

density,, with a particular

section acts as a material with aneffective strength

and density

2)( fBff

2)( fB

Performance index for failure in

bending including shape,

can be written as

3/2

3

)(f

f

BM

2

3/22

3)(

))((f

B

f

BfM

EXAMPLE 1, Failure in bending

EngineeringAlloys

Polymer

Foams

0.1

1

10

100

1000

10000

0.1 1 10 100

Density, (Mg/m3)

Strength,f(MPa)

Ceramics

Composites

Search

Region

Woods

Elastomers

Engineering

Polymers

=1

=10


21/34

Micro-Shape Factors

Material Micro-Shape

+

Macro-Shape,

+

Macro-Shape from

Micro-Shaped Material,

=

Up to now we have only

considered the role of

macroscopic shape on the

performance of fully dense

materials.

However, materials can have

internal shape, Micro-Shapewhich also affects their

performance,

e.g. cellular solids, foams,

honeycombs.


=



22/34

Micro-Shape Factors

Prismatic cells

Concentric cylindrical

shells with foam between

Fibres embedded

in a foam matrix

Consider a solid cylindrical beam expanded, at constantmass, to a circular beam with internal shape (see right).

Stiffness of the solid beam:3

1

L

IECS ooBo

On expanding the beam, its density falls from to ,

and its radius increases from to

o

oo rr

2/1

or

oo

oo IrrI

2

4

2

4

44

The second moment

of area increases to

o

o

EE

If the cells, fibres or rings are

parallel to the axis of the beam then

The stiffness of the

expanded beam is thus

ooB

L

EIC

L

EICS

3

1

3

1Shape Factor:

o

o

e

BS

S


23/34

Function

Tie

Beam

Column

Shaft

Mats. Selection: Multiple Constraints

Objective

Minimum cost

Minimum weight

Maximum stored

energy

Minimum

environmental impact

Constraint

Stiffness

Strength

Fatigue

GeometryMechanicalThermal

Electrical..

Index

2/1

1

EM

Index

3/2

2fM


24/34

Materials for Safe Pressure Vessels

DESIGN REQUIREMENTS

Function Pressure vessel =contain

pressurep

Objective Maximum safety

Constraints (a) Must yield before break

(b) Must leak before break

(c) Wall thickness small toreduce mass and cost

Yield before break

2

2

,

f

IC

CC

IC K

Caa

CK

f

ICKM

1

Leak before break

f

IC

ICC

f

KpRC

t

CKta

pR

tt

pR

22

4

2/2

2,2

f

ICKM

2

2

Minimum strength

f

M 3


25/34

Materials for Safe Pressure Vessels

SearchRegion

M3 = 100 MPa

M1 = 0.6 m1/2

f

ICKM

1f

ICKM

2

2 fM 3

Material M1

(m1/2)

M3

(MPa)

Comment

Tough steels

Tough Cu alloys

Tough Al alloys

Ti-alloys

High strength Al

alloys

GFRP/CFRP

>0.6

>0.6

>0.6

0.2

0.1

0.1

300

120

80

700

500

500

Standard.

OFHC Cu.

1xxx & 3xxx

High strength,

but low safety

margin. Good

for light

vessels.


26/34

1. Express the objective as an equation.

2. Eliminate the free variables using each constraint in turn, giving a set of

performance equations (objective functions) of the form:

where f, gand h are expressions containing

the functional requirements F, geometry M

and materials indices M.

3. If the first constraint is the most restrictive (known as the active constraint)

then the performance is given by P1, and this is maximized by seekingmaterials with the best values ofM1. If the second constraint is the active

one then the performance is given by P2 and this is maximized by seeking

materials with the best values ofM2; and so on.

N.B. For a given Function the Active Constraint will be material dependent.

Multiple Constraints: Formalised

)()()(

)()()(

)()()(

)()()(

33333

22222

11111

iiiii MhGgFfP

MhGgFfP

MhGgFfP

MhGgFfP


27/34

Multiple Constraints: A Simple Analysis

A LIGHT, STIFF, STRONG BEAM The object function is ALm

Constraint 1: Stiffness where so,3

1

L

EICS

12

4tI

2/1

2/5

2/1

1

1

12

EL

C

Sm B

Constraint 2: Strength where so,Ly

ICF

m

ff

2 2

t

ym 3/23/5

3/2

2

2

6

f

f

LC

F

m

If the beam is to meet both constraints then, for a given material, its weight is

determined by the larger ofm1 orm2

or more generally, foriconstraints ).....,,max(~ 321 immmmm

Material E

(GPa)

f

(MPa)

(kgm-3)

m1

(kg)

m2

(kg) (kg)

1020 Steel

6061 Al

Ti 6-4

205

70

115

320

120

950

7850

2700

4400

8.7

5.1

6.5

16.2

10.7

4.4

16.2

10.7

6.5

m~Choose a material

that minimizes m~


28/34

Multiple Constraints: Graphical

log Index M1

log

IndexM

2

Construct a materials selection map based on

Performance Indices instead of materials

properties.

The selection map can be divided into two

domains in each of which one constraint is active.

The Coupling Line separates the domains and iscalculated by coupling the Objective Functions:

where CCis the Coupling Constant.

22

2222

11111

)()(

)()(MCM

GgFf

GgFfM c

Coupling Line

M2 = CCM1

M1 Limited

Domain

M2 Limited

Domain

A

B

Materials with M2/M1>CC , e.g. , are limited by

M1 and constraint 1 is active.

Materials with M2/M1


29/34

Multiple Constraints: Graphical

Coupling Line

M2 = CCM1

Search

Area

C

log Index M1

log

Index

M2

M1 Limited

Domain

M2 Limited

Domain

A

B

C

C

A box shaped Search Region is identified with its

corner on the Coupling Line.Within this Search Region the performance is

maximized whilst simultaneously satisfying both

constraints. are good materials.

M1 Limited

Domain

M2 Limited

Domain

A

B

Coupling Line

M2 = CCM1

log Index M1

log

Index

M2

C

Search

Area

A C

Changing the functional requirements For geometry

G changes CC, which shifts the Coupling Line, altersthe Search Area, and alters the scope of materials

selection.

Now and are selectable.


30/34

Windings for High Field Magnets

DESIGN REQUIREMENTS

Function Magnet windings

Objective Maximize magnetic field

Constraints (a) No mechanical failure(b) Temperature rise


31/34


),(f

L

NiB foThe field (weber/m2) is

where o = the permeability of air, N= number of turns, i= current,f= filling factor,

f(,) = geometric constant, = 1+(d/r), = L/2r

CONSTRAINT 1: Mechanical Failure

Radial pressure created by the field

generates a stress in the coil

),(f2

2

o

Bp

d

rB

d

pr

o ),(f2

2

must be less than the yield stress of the coil material y

and hence 2/1),(f2

r

dB yofailure

So, Bfailure is maximized

by maximizingyM 1


32/34


CONSTRAINT 1: Overheating

So, Bheat is maximized

by maximizing

e

pCM

2

The energy of the pulse is (Re = average of the resistance over the

heating cycle, tpulse = length of the pulse) causes the temperature of the coil to rise by

where e

= electrical resistivity of the coil material

Cp = specific heat capacity of the coil material

pulseetRi2

po

pulsee

Cd

tBT

22

2

If the upper limit for the change in temperature

is Tmaxand the geometric constant of the coilis included then the second limit on the field is ),(f

2/1

max

22

epulse

po

heat

t

TCdB


33/34


Material

y

(MPa)

(Mg/m3)

Cp

(J/kgK)

e

(10-8m)

Bfailure

(wb/m2)

Bheat

(wb/m2) (wb/m2)

High conductivity Cu

Cu-15%Nb compositeHSLA steel

250

7801600

8.94

8.907.85

385

368450

1.7

2.425

35

6289

113

9230

35

6230

Pulse length = 10 ms

B~

),min(~ heatfailure BBB In this case the field is limited by the lowest ofBfailure and Bheat: e.g.

),(f

2/1

max

22

epulse

po

heatt

TCdB

2/1),(f2

r

dB yofailure

2212

),(fMCM

t

TrdM C

pulse

maxfo

Thus defining the Coupling Line


34/34

10

100

1000

10000

100 1000 10000

Index M2 (10-8

m3/J)

IndexM1(MP

a)

Search Region:

Ultra-short pulse

Search Region:long pulse

Search Region:

short pulse

HSLA steels

CuAl-S150.1

Cu-4Sn

Cu-Be-Co-Ni

Be-Coppers

GP coppers

HC Coppers

Cu-NbCu-Al2O3

Cu-Zr


Material Comment

Continuous and long

pulse

High purity coppers

Pure Silver

Short pulse

Cu-Al2O3 composites

H-C Cu-Cd alloysH-C Cu-Zr alloys

H-C Cu-Cr alloys

Drawn Cu-Nb comps

Ultra short pulse,

ultra high field

Cu-Be-Co-Ni alloys

HSLA steels

Best choice for low field,

long pulse magnets (heat

limited)

Best choice for high field,

short pulse magnets (heatand strength limited)

Best choice for high field,

short pulse magnets

(strength limited)

e

pCM

2yM 1

Documents

Design Notes3 (6)