MECH4301 2008 Lecture 6 (1/3) Shape Factors 1/23 Material and Shape: Textbook Chapters 11 and 12...

MECH4301 2008 Lecture 6 (1/3) Shape

Factors 1/23

Material and Shape:

Textbook Chapters 11 and 12

Lecture 6 (1/3)

Efficient?

Materials for efficient structuresTute # 3, 6 exercises, 3 afternoons to solve

them. Read the instructions (BB) carefully. Due Mo

Sept. 15, 4:00 pm

MECH4301 2008 Materials Selection in Mechanical Design

Laboratory Group II: Meet on Friday Au 29, 8 am, room 413, Bldng 43.

“Efficient” = use least amount of

material for given stiffness

or strength.

To create a deformation work-stress chart for foams, use (densification strain * yield strength) as y-axis and yield strength as x-axis.) See Announcement in Bb.


Factors 2/23

Shape and mechanical efficiency

• Section shape becomes important when materials are loaded in bending, in torsion, or are used as slender columns.

• Examples of “Shape”:

• Shapes to which a material can be formed are limited by the material itself.

Shapes from: http://www.efunda.com/math/areas/RolledSteelBeamsS.cfm

Is shape important for

tie rods?


Factors 3/23

Shape and mode of loadingStandard structural members

Loading: tension/compression

Area A and shape IXX, IYY matter

Area A and shape J matter

Area A and shape Imin matter

Area A matters,not shape

Loading: bending

Loading: torsion

Loading: axial compression

Certain materials can be made to certain shapes: what is the best material/shape combination (for each loading mode)?


Factors 4/23

Shape efficiency: bending stiffness pp. 289-290

1212

24

oo

AbI

b

b

Area A is constant

Area Ao = b2

modulus E unchanged

Neutral reference section

Shaped sections

Define a standard reference section: a solid square, area A = b2

30

L

IECSo

3

L

IECS

31

L

EICFS

Moments of Sections; p 477

12

3bhI

221212

AAE

E

S

S

oooo

e

IIII

II

Ao = A

Define shape factor for elastic bending,

measuring efficiency, as


Factors 5/23

A shaped beam of shape factor for elastic bending, e = 10, is 10 times stiffer than a solid square section

beam of similar cross section area.

212

AE

E

S

S

ooo

e

III

II

bending stiffness


Factors 6/23

I-sections

Properties of the shape factor The shape factor is dimensionless -- a pure number.

It characterises shape, regardless of size.

Circular tubes

10e

10e

These sections are φe times stiffer in bending than a solid square section of the same cross-sectional area

Increasing size at constant shape = constant SF

Rectangular Sections e = 2


Factors 7/23

Define a standard reference section: a solid square, area A = b2

Shape efficiency: bending strength p. 294/5

66

2/3

0

3 Ab

y

IZ

m

o

o

** Zy

IM

mf

maxy

ZI

b

b

Area A is constant

Area A = b2

yield strength unchanged

*

Neutral reference section

Moments of Sections; p 477

my

IZ ,modulussection

• Define shape factor for the onset of plasticity (failure),

measuring efficiency, as

** Zy

IM

m

f

*

0

*0

0 Z

y

IM

m

f

2/3*

*

6A

Z

Z

Z

Z

Z

M

M

oofo

ff

A = Ao


Factors 8/23

A shaped beam of shape factor for bending strength, f = 10, is 10 times stronger than a solid square section

beam of similar cross section area.

2/3*

*

6A

Z

Z

Z

Z

Z

M

M

oofo

ff

bending strength


Factors 9/23

Section shape Area Am

Secondmoment I, m4

Elastic shapefactor

bh12

hb 3b

h

ab ba4

3b

a3

tr2

)rr( 2i

2o

tr

)rr(4

3

4i

4o

)tr(

t

r3

)tb,h(

)bh(t2

)

h

b31(th

6

1 3 tb,h(

)h/b1(

)h/b31(

t

h

2

12

)tb,h(

tb2

)hh(b io

2o

3i

3o

htb2

1

)hh(12

b

)tb,h(

tb

h

2

3 2o

)tb,h(

)bh(t2

)

h

b31(th

6

1 3

)tb,h(

)h/b1(

)h/b31(

t

h

2

12

h

t

2t

b

h

b

t

2a

2b

h

b

b

t

hohi

2ro2ri

t

Tabulation of shape factors (elastic bending) p. 292/3

t

r

t

r

rt

tr

Ao

e

3

)2(1212

2

3

2 I

II

b

h

hb

bh

Ao

e

22

3

2

1

121212

III

A2 = Ao2

Second moment of section, I


Factors 10/23

Comparison of shapes done so far at constant material (E, y) and given cross section area, A

How to compare different materials and different shapes at:

Constant structural stiffness, S ?

Constant failure moment, Mf ?

Material substitution at constant stiffness or strength allowing for differences in shape


Factors 11/23

L

mA

A

Ae

o

e

12 I 12

I

I 2

2

I

m = massA = areaL = length = densityb = edge lengthS = stiffnessI = second moment of areaE = Youngs Modulus

Beam (shaped section).

Bending stiffness of the beam S:

Trick to bring the Shape Factor in ?

Eliminating A from the eq. for the mass gives:

3L

IECS

2/1

2/1512

EC

LSm

e

LAm

Chose materials with largest

2/1Ee

Minimise mass, m, where:

Function

Objective

ConstraintL

FArea A

Shape factor part of the material index

Indices that include shape (1): minimise mass at constant stiffness p. 310


Factors 12/23

Indices that include shape (2): minimise mass at constant strength p . 311

L

mA

A

Af

of

6 Z

Z6

Z

Z2/3

2/3

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

Beam (shaped section).

Bending strength of the beam Mf:

Trick to bring the Shape Factor in ?

Eliminating A from the equation for m gives:

** Zy

IM

mf

3/2*

3/26

f

f LMm

LAm

Chose materials with largest

3/2*f

Minimise mass, m, where:

Function

Objective

ConstraintL

FArea A

Shape factor part of the material index


Factors 13/23

From Lecture 4: Demystifying Material Indices (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 determined by the ratio of material indices.

Same conclusion applies to bending strength.

Unshaped mass, Material 1

Unshaped mass Material 2


Factors 14/23

Demystifying Shape Factors (elastic bending)

2/1

1

1

2/1512

EC

LSmo

2/1

1

1

2/1512

EC

LSm

e

s

2/11

2/11

2/11

1 1

)(

E

Em

m

o

s

Shaping (material fixed) at constant bending stiffness

reduces the mass of the component in proportion to e

-1/2 .

Optimum approach: simultaneously maximise

both M and .

Unshaped mass

Shaped mass, same material, same S

Q: Is the cross section area constant when going

from mo to ms?


Factors 15/23

Demystifying Shape Factors (failure of beams)

3/2*

3/26

LMm fo 3/2*

3/26

f

fs LMm

3/23/2*

3/2* 1

ffo

s

m

m

Unshaped mass

Shaped mass, same material, same Mf

Shaping (material fixed) at constant bending strength

reduces the mass of the component in proportion to f

-

2/3.Optimum approach:

simultaneously maximise both M and .

EXAM QUESTION: Is the cross section area constant when going

from mo to ms?


Factors 16/23

Material , Mg/m3 E, GPa e,max

1020 Steel 7.85 205 65 1.8 14.7

6061 Al 2.70 70 44 3.1 20.4

GFRP 1.75 28 39 2.9 18.9

Wood (oak) 0.9 13 8 4 11.4

/2/1E /ρE 1/2maxe,

Practical examples of material-shape combinations

/2/1E

• Materials for stiff beams of minimum weight

• Fixed shape (e fixed): choose materials with greatest

• Shape e a variable: choose materials with greatest

Same shape for all (up to e = 8): wood is best

Maximum shape factor (e = e,max): Al-alloy is best

Steel recovers some performance through high e,max

/ρE 1/2maxe,


Factors 17/23Density (typical) (Mg/m^3)

0.01 0.1 1 10

Yo

un

g's

Mo

du

lus

(typ

ica

l) (

GP

a)

1e-004

1e-003

0.01

0.1

1

10

100

1000

Concrete

Titanium

Cork

PP

Flexible Polymer Foams

Rigid Polymer Foams

Tungsten Carbides

Steels Nickel alloys

Copper alloys

Zinc alloys

Lead alloys

Silicon CarbideAluminaBoron Carbide

Silicon

Al alloys

Mg alloys

CFRP

GFRPBamboo

Wood

Plywood PET

PTFE

PE

PUR PVC

EVA

Silicone

Polyurethane

Neoprene

Butyl Rubber

Polyisoprene

CE

2/1

Note that new material with

Shape on selection charts: stiffness p. 312/3

Al: e = 44

Al: e = 1

Density (Mg/m3)

You

ng’s

mod

ulus

(G

Pa)

e

e

e

e

e

e EE

E

/

/

/

2/1

2/1

2/1

e

s /

es EE /



Factors 18/23

Shape on selection charts: stiffness p. 314

Density (Mg/m^3)0.1 1 10

Yo

un

g's

Mo

du

lus

(GP

a)

1

10

100

1000

Al Sf=44

Bamboo SF=1

Steel SF = 65

Bamboo SF =5.6

steel SF=1

Al SF =1

Drag the labels along lines of slope 1

Selection line of slope 2

Unshaped Steel SF =1

Unshaped Aluminium

Unshaped Bamboo SF= 1

Shaped aluminium SF = 44

Shaping makes Steel competitive with Al and Bamboo

Shaped Bamboo SF=5.6

Shaped steel SF=65


Factors 19/23

Note that new material with

2

3/22*

2

2

3/2*

3/2*

/

/

/f

f

f

f

f

f

2** / f

s 2/ f

s

Shape on selection charts: strength p. 314

3/2*

100 2 fsteel

1 2 fsteel



Factors 20/23

Shape on selection charts: strength p. 314

Density (Mg/m^3)0.01 0.1 1 10

Te

nsi

le S

tre

ng

th (

MP

a)

10

100

1000

Bamboo SF =1

Al 2024, SF=10 SF^2=100

steels SF=7 SF^2=49

bamboo SF = 2 SF^2=4

steel SF =1

Al Sf =1

selection line slope 1.5Selection line of slope 1.5

Shaped Steel SF=7; (SF)2=49

Shaped Bamboo SF=2 (SF)2=4

Shaping makes Steel competitive with Al and Bamboo

Shaped Aluminium SF=10; (SF)2=100


Factors 21/23

Shaping at constant cross section A increases the bending stiffness or strength by at constant mass. This stems from the definition of shape factor e = S/So= I/Io f = M/Mo = Z/Zo

Dragging the labels in the CES charts is equivalent to shaping at constant bending stiffness or strength, so the mass is reduced by 1/e

1/2 (stiffness) or

by 1/f2/3 (strength).

Exam question: (to get everybody confused!)


Factors 22/23

Examples of indices including shape p. 318

Same as elastic bending


Factors 23/23

This afternoon: solve Exercises E8.1, 8.8, 8.9 and 8.12.

Leave 8.6 / 8.7 for next sessions.

-Tutorial 3 (E8. Materials and Shape) (6 Exercises). Solve in this order: E8.1; E8.8; E8.9; E8.12; (solve either E8.6 or E8.7) (see

hints and instructions in BB). Exercise #6 for Tute 3: Show that the shape factors of Table 12.5 (p.

325) are a factor 4/3 = 1.33 too large.


Factors 24/23

Example using CES: dragging labels


Factors 25/23

End of Lecture 6

two more lectures re. shape factors to follow


Factors 26/23

Shape factors for twisting and buckling

2T2

0o

T

A

K14.7 14.0K

K

K

S

S A

ToT

3/2A

Q8.4

Q

Q

ofT

Failure under torsion p. 296

Buckling p. 296

212

AE

E

S

S

ooe

III

Same as elastic bending

Elastic twisting p. 294Moments of Sections; p 477Moments of Sections; p 477

MECH4301 2008 Lecture 6 (1/3) Shape Factors 1/23 Material and Shape: Textbook Chapters 11 and 12...

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Transcript of MECH4301 2008 Lecture 6 (1/3) Shape Factors 1/23 Material and Shape: Textbook Chapters 11 and 12...