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2 - Ashby Method

2.7 - Materials selection and shape

Outline

• Shape efficiency

• The shape factor, and shape limits

• Material indices that include shape

• Graphical ways of dealing with shape

Resources:

• M. F. Ashby, “Materials Selection in Mechanical Design” Butterworth Heinemann, 1999

Chapter 7

• W. C. Young, R. G. Budynas, “Roark’s Formulas for Stress and Strain” 7th ed, McGraw-Hill, 2002

• The Cambridge Material Selector (CES) software -- Granta Design, Cambridge

(www.grantadesign.com)

Structural components

Moments of area

∫=A

2xx dA yI

∫=A

2yy dA xI

∫=A

2 dA r J

Moment of area about axis x

Moment of area about axis y

Polar moment of area

dA

They depend on shapes

Modes of loading: Axial loading

F

Strain L

ε

Stress A

Fσ

δ=

=

AE

F

E

σε Eεσ ===

• From Hooke’s Law (linearly elastic material):

• From the definition of strain:

Lε

δ=

AE

FL=δ

Stiffness L

AE

FS ==

δ

σσσσ

⇒

⇒

Modes of loading: Bending

Pure Bending: Prismatic members subjected to couples acting in the longitudinal plane crossing one of the principal inertia axes

After deformation, the length of the neutral surface remains L. At other sections:

( )( )

maxz

maxmax

z

εc

yε

ε

cρ

ρ

cε

linearly) (varies Strain ρ

y

ρ

y

Lε

yρyρL L'

yρL

−=

==

−=−==

−=−−=−=

−=′

θ

θδ

θθθδ

θ

⇒

x

z


For a linearly elastic material:

linearly) (varies Stress σc

y

Eεc

yEεσ

max

maxzz

−=

−==

/cI

M

Z

Mσ

xxmax == (c = ymax)

/yI

Mσ

xxz =

σz

IXX = moment of area

about the bending axis

modulus strength Bending c

IZ xx=


3xx

L

IECFS ==

δ

IXX = moment of area about the bending axis

C = constant (depending on the loading conditions)

StiffnessF

δ

( )

= zM

dz

ydI E

2

2

xx

z

Modes of loading: Torsion

Torsion: Prismatic members subjected to twisting couples or torquesT

Consider an interior section of the shaft. As a torsional load is applied, the shear strain is equal to angle of twist.

Shear strain ( twist angle and radius)

maxmaxc

ρ

L

cγγγ ==

φ

L

ρ ρL'AA

φφ === γγ

))

⇒

∝


T

For a linearly elastic material:

linearly) (varies stress Shear c

ρ

Gc

ρG

max

max

τ

γγτ

=

==

K/c

T

Q

Tmax ==τ (c = ρmax)

K/ρ

T=τ K = torsional moment of area

modulus strength Twisting c

KQ =


T

J/c

T

Q

Tmax ==τ (c = ρmax)

J/ρ

T=τ J = polar moment of area

L

JGTST ==

φStiffness

• Cross-sections of noncircular (non-axisymmetric)

shafts are distorted when subjected to torsion.

• Cross-sections for hollow and solid circular shafts

remain plain and undistorted because a circular

shaft is axisymmetric.

K = J for circular sections only

L

KGTST ==

φStiffness

===

K

Tρ

ρG

L

Gρ

L

ρ

L τγφ

Modes of loading: Buckling

Buckling: Prismatic members subjected to compression in unstable equilibrium

• In the design of columns, cross-sectional area is selected such that

- allowable stress is not exceeded

yσA

Fσ ≤=

- deformation falls within specifications

limAE

FLδδ ≤=

• After these design calculations, may discover that the column is unstable under loading and that it suddenly buckles.

FF


F F

F’

F’

• Consider ideal model with two rods and torsional spring. After a small perturbation

( )

moment ingdestabiliz ∆2

LFsin∆

2

LF

moment restoring 2∆k

==

=

θθ

θ

• Column is stable (tends to return to aligned orientation) if

( )

L

4kFF

2∆k∆2

LF

cr =<

< θθ


F

The critical loading is calculated from Euler’s formula

( )

2

2

cr

2

2

cr

2

min2

cr

Eσ

rL

Eσ

L

EIF

λ

π

π

π

=

=

=

) radius inertia/A Ir ( min2 =

) sslendernes /rL ( 222 =λ

Stress corresponding to critical loadingL


2e

min2

crL

EIF

π=

Le = Equivalent length (length of free inflexion, distance between two subsequent inflexion points)

FF F

F

Shape efficiency

“Shape” = cross section formed to atubes

I-sections

hollow box-sectionsandwich panels

“Efficient” = use least material for given stiffness or strength

Shapes to which a material can be formed are limited by the material itself (processability and mechanical behaviour)

Goals: - quantify the efficiency of shape

- understand the limits to shape

- develop methods for co-selecting material and shape

Certain materials can be made to certain shapes: what is the best combination?

Shape and mode of loading

Area A matters,not shape

Area A and shape

(IXX) matter

Area A and shape (J, K) matter

Area A and shape

(Imin) matter

When materials are loaded in bending, in torsion, or are used as columns, section shape becomes important


Tie-rod

Minimise mass m:

m = A L ρρρρ

Function

Objective

Constraints

m = mass

A = area

L = length

ρ = density

S = stiffness

E = Youngs Modulus

Stiffness of the tie S:

L

AES =

Area A matters, not shape

L

FF

Area A


Tie-rod

Minimise mass m:

m = A L ρρρρ

L

FF

Area A

Must not fail under load F:

Function

Objective

Constraints

m = mass

A = area

L = length

ρ = density= yield strength

yσ

F/A < σσσσy

Area A matters, not shape


m = mass

A = area

L = length

ρ = densityb = edge length

S = stiffness

I = second moment of area

E = Youngs Modulus

Beam (solid square section).

Stiffness of the beam S:

I is the second moment of area:

3L

IECS =

12

bI

4

=

ρ=ρ= LbLAm 2

b

b

L

F

Minimise mass, m, where:

Function

Objective

Constraint

Area A and shape matter


m = mass

A = area

L = length

ρ = densityb = edge length


σy = yield strength

Beam (solid square section).

Must not fail under load F

I

b/2M

Z

Mσ y

⋅=>

ρ=ρ= LbLAm 2

b

b

L

F


Function

Objective

Constraint

I is the second moment of area:12

bI

4

=

Area A and shape matter


Definition of Shape Factor

� Bending has its “best” shape: beams with hollow-box or I-sections

are better than solid sections of the same cross-sectional area

� Torsion too has its “best” shape: circular tubes are better than

either solid sections or I-sections of the same cross-sectional area

To characterize this we need a metric - the shape factor – a way of measuring

the structural efficiency of a section shape

- specific for each mode of loading

- independent of the material of which the component is made

- dimensionless (regardless of shape scale)

We define shape factor the ratio of the stiffness (or strength) of the

shaped section to the stiffness (or strength) of a ‘reference shape’, with

the same cross-sectional area (and thus the same mass per unit length)

Shape efficiency: Bending stiffness

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

(alternatively: solid circular section)

• Second moment of area is I; stiffness scales as EI (S = CEI/L3)

� Take ratio of bending stiffness S of shaped section to that (So) of

a neutral reference section of the same cross-section area

b

b

L

F

3L

IECS =


2oo

eB

A

I12

IE

IE

S

S ===φ

12

A

12

b 24

o ==I

Define shape factor for elastic bending, measuring efficiency, as

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

(alternatively: solid circular section)

• Second moment of area is I; stiffness scales as EI (S = CEI/L3)

� Take ratio of bending stiffness S of shaped section to that (So) of

a neutral reference section of the same cross-section area

b

b

Area A is constant

Area A = b2 Area A and modulus E unchanged


Properties of Shape Factor

� The shape factor is dimensionless -- a pure number

� It characterizes shape

Each of these is roughly 2-10-12 times stiffer in bending than a solid square section of the same cross-sectional area

Increasing size at constant shape

Tabulation of Shape Factors

(standard reference section:

solid square section)

2oo

eB

A

I12

IE

IE

S

S ===φ

12

A

12

bI

24

o ==


(standard reference section:

solid circular section)

2oo

eB

A

I4

IE

IE

S

S π===φ

π

π

4

A

4

r I

24

o ==

Shape efficiency: Bending strength

3/2oy

y

fo

ffB

A

Z6

Z

Z

M

M ===

σ

σφ

6

A

6

b

b

2

12

bZ

3/234

o==⋅=

maxyZ

I=

Define shape factor for failure in bending, measuring efficiency, as

• Take ratio of bending strength (failure moment) Mf of shaped section to that (Mf,o) of a reference section (solid square) of the same cross-section area

• Section modulus for bending is Z; strength (Mf) scales as (Mf = σy Z)Zyσ

Area A = b2

Area A and yield strengthunchanged

yσ

Area A is constant

b

b

Shape efficiency: Bending strength

Shape efficiency: Twisting stiffness

2ooT,

TeT

A

K7,14

GK

KG

S

S ===φ

23

o A0,14h

b0,581

3

hbK =

−⋅

⋅=

Define shape factor for elastic twisting, measuring efficiency, as

• Torsional moment of area is K (= J for circular sections); stiffness scales as KG

� Take ratio of twisting stiffness ST of shaped section to that (ST,o) of a

reference section (solid square) of the same cross-section area

b

b

b = h

Area A = b2 Area A and modulus G unchanged

L

KG

θ

TST ==

Shape efficiency: Twisting stiffness

Shape efficiency: Twisting strength

3/2oof,

ffT

A

Q4,8

Q

Q

T

T ===

τ

τφ

4,8

A

4,8

b

1,8b3h

hbQ

3/2322

o ==+

=

maxr

JQ =

Define shape factor for failure in twisting, measuring efficiency, as

• Take ratio of twisting strength (failure torque) Tf of shaped section to that (Tf,o) of a reference section (solid square) of the same cross-section area

• Section modulus for twisting is Q; strength (Tf) scales as (Tf = τ Q)Q τ

Area A and strengthunchanged

τ

b

b

(for circular sections only)

b = h

Area A = b2

Shape efficiency: Twisting strength

Shape efficiency: Resistance to buckling

• Take ratio of critical load (Euler load) Fcr of shaped section to that (Fcr,o) of a reference section (solid square) of the same cross-section area

• Critical load (Fcr) scales as (Fcr = π2EImin/Le2)

• The shape factor is the same as that for elastic bending ( ), with I replaced by Imin

minEI

2min

omin,

min

ocr,

crBck

A

I12

IE

IE

F

F ===φ

12

A

12

bII

24

oomin, ===

Define shape factor for resistance to buckling, measuring efficiency, as

b

b

Area A = b2

Area A and modulus E unchanged

eB φ


Limits for Shape Factors

If you wish to make stiff, strong structures that are efficient (using as little material as possible) then make shapes with shape factors as large as possible

Two types of limit for shape factors

- manufacturing constraints (processability of materials)

- mechanical stability of shaped sections

Limits for Shape Factors

If you wish to make stiff, strong structures that are efficient (using as little material as possible) then make shapes with shape factors as large as possible

Theoretical limit:y

eB

E2.3

σ≈φ

Modulus

Yield strength

In seeking greater efficiency, a shape is chosen that raises the load required for the simple failure modes (yield, fracture).But in doing so, the structure is pushed nearer the load at which new failure modes become dominant.

Two types of limit for shape factors

- manufacturing constraints (processability of materials)

- mechanical stability of shaped sections

Local buckling

≈ e

BfB φφ

What values of φφφφBe exist in reality?

⇒=φ2

eB

A

I12

( ) ( )

φ+=

12

logA2logIlog

eB

Slope = 2

x

z

3

xx

L

IECS =

x

Ixx > Ixx > Ixx

What values of φφφφBf exist in reality?

⇒=φ3/2

fB

A

Z6

( ) ( )

φ+=

6

logAlog

2

3Zlog

fB

Slope = 3/2

x

z

max

xx

y

IZ =

x

Ixx > Ixx > Ixx

What values of φφφφBe exist in reality?

12log)A(log2)(log

A

12 e2e

ϕ+=⇒=ϕ I

I

Section Area, A (m^2)1e-005 1e-004 1e-003 0.01 0.1

Se

cond

Mom

en

t of A

rea (

majo

r), I_

ma

x (m

^4

)

1e-011

1e-010

1e-009

1e-008

1e-007

1e-006

1e-005

1e-004

1e-003

0.01

Extruded Al-tube

Extruded Al-angle

Pultruded GFRP tube

Pultruded GFRP I-section

Steel Universal Beam

Pultruded GFRP Channel

Steel tube

Glulam rectangular

Pultruded GFRP Angle

Softwood rectangular

Extruded Al A-angle

Steel tube

Extruded Al I-section

Extruded Al-Channel

Second m

om

ent of are

a, I (m

4)

Section Area, A (m2)

Data for structural steel, 6061 aluminium, pultruded GFRP and wood

100e=ϕ

Slope = 2

1e=ϕ

φBe = ϕe

Indices that include shape

m = mass

A = areaL = length

ρ = densityb = edge lengthS = stiffness


E = Youngs Modulus

Beam (shaped section)

Bending stiffness of the beam S:

I is the second moment of area:

Combining the equations gives:

3L

IECS =

( )

ϕ

ρ

=

2/1e

2/15

EC

LS12m

ρ= LAm

Chose materials with smallest( )

ϕ

ρ2/1

eE


Function

Objective

Constraint

2/1

e2e

12A

A12

ϕ==ϕ

II

L

FArea A

CE

SLI

3

=

Material ρ, Mg/m3 E, GPa ϕe,max

1020 Steel 7.85 205 65 0.55 0.068

6061 T4 Al 2.70 70 44 0.32 0.049

GFRP 1.75 28 39 0.35 0.053

Wood (oak) 0.9 13 8 0.25 0.088

2/1E/ρ ( ) 2/1max,e E/ ϕρ

Selecting material-shape combinations

Materials for stiff, shaped beams of minimum weight

• Fixed shape (ϕe fixed): choose materials with low

• Shape ϕe a variable: choose materials and shapes with low

• Commentary: Fixed shape (up to ϕe = 8): wood is best

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

Steel recovers some performance through high ϕe,max

2/1E

ρ

( ) 2/1eEϕ

ρ


⇒=φ2

eB

A

I12

( ) ( )

φ+=

12

logA2logIlog

eB


⇒= I ES L

( ) ( ) ( )LS logIlog-Elog +=


⇒ρ= Am/L

( ) ( ) ( )m/L loglog-Alog +ρ=


Required section stiffness:

EI = 106 N.m2

Shape factor:

φBe = 10



EI = 106 N.m2

Shape factor:

φBe = 10



EI = 106 N.m2

Shape factor:

φBe = 2



EI = 106 N.m2

Shape factor:

φBe = 30


Required section strength:

σyZ > Vmin


Required section strength:

σyZ > Vmin


Selection with fixed shape


Shape

S

Selection with variable shape


four

4 S

• When the groups are separable, the optimum choice of materialand shape becomes independent of the detail of the design.It is the same for all geometries G and all values of functionalrequirements F.

• The performance for all F and G is maximized by maximizingf3(M) and f4(S).


four

4 S

• In theory f4(S) is independent of the material (shape factors dependon shape only).

• In reality the shape factors depend on material (because of constraintsfrom material-process-shape relations, and limits from processabilityand mechanical behaviour of material which form the shape), thereforef3(M) f4(S) constitutes the new performance index. .

• Shaped material can be considered as a new material with modified (improved) properties.

Shape on selection charts

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

Young's

Modulu

s (ty

pic

al) (

GP

a)

1e-004

1e-003

0.0 1

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 Carbide

AluminaBoron 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

=ρ

Al: ϕe = 1

Density (Mg/m3)

You

ng

’s m

odulu

s (

GP

a)

( ) ( ) ( )1/21/2

e

e

1/2

e*E

*

E/

/

E

ρρρ=

ϕ

ϕ=

ϕNote that New material with

e/* ϕρ=ρ

e/E*E ϕ=

Shape on selection charts

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

Young's

Modulu

s (ty

pic

al) (

GP

a)

1e-004

1e-003

0.0 1

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 Carbide

AluminaBoron 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

=ρ

Al: ϕe = 44

Al: ϕe = 1

Density (Mg/m3)

You

ng

’s m

odulu

s (

GP

a)

ρAl /44

EAl /44

( ) ( ) ( )1/21/2

e

e

1/2

e*E

*

E/

/

E

ρρρ=

ϕ

ϕ=

ϕNote that New material with

e/* ϕρ=ρ

e/E*E ϕ=

Data organisation: Structural sections

Kingdom Family AttributesMaterial and Member

• Angles

• Channels

• I-sections

• Rectangular

• T-sections

• Tubes

Extruded Al alloy

Pultruded GFRP

Structural steel

Softwood

Structural sections

A record

Material properties

, E,

Dimensions A ...

Section properties:

I, Z, K, Q ...

Structural properties:

EI, Z, GK ...yσ

yσρ

Standard

prismatic sections

Part of a record for a structural section

Material propertiesPrice 3.99 - 4.87 GBP/kg

Density 1.65 - 1.75 Mg/m^3

Young's Modulus17 - 18 GPa

Yield Strength 195 - 210 MPa

DimensionsDiameter, B 0.0439 - 0.0450 m

Thickness, t 2.54e-3 - 3.81e-3 m

Section propertiesSection Area, A 3.3e-004 - 4.93e-004 m^2Second Moment of Area (maj.), I_max 7.11e-008 - 1.05e-007 m^4

Second Moment of Area (min.), I_min 7.11e-008 - 1.05e-007 m^4

Section Modulus (major), Z_max 3.23e-006 - 4.68e-006 m^3

Section Modulus (minor), Z_min 3.23e-006 - 4.68e-006 m^3

Etc.

Structural propertiesMass per unit length, m/l 0.562- 0.837kg/m

Bending Stiffness (major), E.I_max 1230 - 1810 N.m^2

Bending Stiffness (minor), E.I_min 1230 - 1810 N.m^2

Failure Moment (major), Y. Z_max 647 - 935 N.m

Failure Moment (minor), Y. Z_min 647 - 935 N.mEtc.

Pultruded GFRP Vinyl Ester (44 x 3.18)

Example: Selection of a beam

ma = mass/unit length

Ca = cost/unit length

D = beam depth

B = width


E = Young’s modulus

Z = section modulus

σy = yield strength

Beam

Required stiffness:

EImax > 105 N.m2

Required strength:

σyZ > 103 N.m

Dimension

B < 100 mm

D < 200 mm

(a) Find lightest beam

(b) Find cheapest beam

Function

Objectives

Constraint L

FB x DSpecification

Applying constraints with a limit stage

Dimensions Minimum Maximum

Depth D mm

Width B mm

Section attributes

Bending Stiffness E.I N.m2

Failure Moment Y. Z N.m

200

100

100,000

1000

Optimisation: Minimising mass/length

Mass per unit length, m/l (kg/m)0.1 1 10 100 1000

Be

nd

ing

Stiffn

ess (

ma

jor)

, E

.I_m

ax (

N.m

^2)

1

10

100

1000

10000

100000

1e+006

1e+007

1e+008

1e+009

Pultruded GFRP tube

Steel tubeExtruded Al I-section

Extruded Al-tube

Steel Universal Beam

Steel Rect.Hollow

Steel Equal Angle

Extruded Al Angle

Bendin

g S

tiffness E

.Im

ax

(Nm

2)

Mass per unit length (kg/m)

Bending Stiffness EI vs.

mass per unit length

E.Imax = 105 Nm2

Selection

box

Results: Selection of a beam

OUTPUT: objective – minimum weight

OUTPUT: objective – minimum cost

Extruded aluminium box section, YS 255 MPa (125 x 56 x 3.0 mm)




Sawn softwood, rectangular section (150 x 36)




The main points

� When materials carry bending, torsion or axial compression, the

section shape becomes important.

� The “shape efficiency” quantify the amount of material needed to carry the load. It is measured by the shape factor, φ.

� If two materials have the same shape, the standard indices for

bending (eg ) guide the choice.

� If materials can be made -- or are available -- in different shapes, then indices which include the shape (eg ) guide the

choice.

� The CES Structural Sections database allows standard sections to

be explored and selected.

2/1E/ρ

( ) 2/1/ Eφρ

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