Download Unit 9. Material and shape: materials for efficient structures

Unit 9.
Material and shape:
materials for efficient
structures
Mike Ashby
Department of Engineering
University of Cambridge
© M. F. Ashby, 2013
For reproduction guidance see back page
This lecture unit is part of a set created by Mike Ashby to help introduce students to materials, processes and rational selection.
The Teaching Resources website aims to support teaching of materials-related courses in Design, Engineering and Science.
Resources come in various formats and are aimed primarily at undergraduate education.
Some of the resources are open access and students can access them. Others are only available to educators using CES EduPack.
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Outline, outcomes and resources
Efficient shapes: tubes, I-beams etc
The shape factor and shape limits
Material indices that include shape
Outcomes
Ability to quantify shape
efficiency
Ability to select efficient
material-shape combinations
Graphical ways of dealing with shape
The CES EduPack Structural Sections database
Resources
“Materials Selection in Mechanical Design”, 4rd edition, Chapters 9 and 10.
Mike Ashby, 2013
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Shape efficiency
When materials are loaded in bending, in torsion, or are used as slender
columns, section shape becomes important
”Shape” = cross section formed to a
tubes
I-sections
tubes
hollow box-section
sandwich panels
ribbed panels
“Efficient” = use least material for given stiffness or strength
Shapes to which a material can be formed are limited by the material itself
Goals:
understand the limits to shape
develop methods for co-selecting material and shape
Mike Ashby, 2013
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Shape and mode of loading
Standard structural members
Tie
Beam
Torsion bar
Column
Area A matters,
not shape
Area A and shape IXX, IYY
matter
Area A and shape J
matter
Area A and shape Imin
matter
Certain materials can be made to certain shapes: what is the best combination?
Mike Ashby, 2013
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Shape efficiency: bending stiffness
Take ratio of bending stiffness S of shaped section to that (So) of a neutral
reference section of the same cross-section area
Define a standard reference section: a solid square with area A = b2
Second moment of area is I; stiffness scales as EI.
4
Io =
Area A is
constant
2
b
A
=
12
12
I = ∫ y 2 b( y ) dy
Area A and
modulus E
unchanged
Area A = b2
b
b
Define shape factor for elastic bending, measuring efficiency, as
ϕe =
Mike Ashby, 2013
S
EI
I
=
= 12 2
So
E Io
A
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Properties of the shape factor
The shape factor is dimensionless -- a pure number.
It characterizes shape.
I-sections,
ϕe ≈ 10
Circular tubes,
ϕe ≈ 10
Increasing size at constant shape
Each of these is roughly 10 times stiffer in bending than a solid
square section of the same cross-sectional area
Mike Ashby, 2013
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Shape efficiency: bending strength
Take ratio of bending strength Ff of shaped section to that (Ff,o) of a
neutral reference section of the same cross-section area
Section modulus of area is Z; strength scales as σ y Z
b3
A3 / 2
Zo =
=
6
6
Area A is
constant
Z =
I
y max
Area A = b2
Area A and yield
strength
σ y unchanged
b
b
Define shape factor for onset of plasticity (failure), measuring efficiency, as
σyZ
Z
Ff
ϕf =
=
= 6 3/2
Ffo
σ y Zo
A
Mike Ashby, 2013
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Tabulation of shape factors
Section shape
Area A
m
h
bh
Second
moment I, m 4
Elastic shape
factor
b h3
12
h
b
π 3
a b
4
3a
πb
π 4 4
( ro − ri )
4
3
π
(r
b
2a
πab
2b
t
2ri
π ( ro2 − ri2 )
2ro
≈ 2πr t
≈ π r3 t
t
h
2 t (h + b)
( h , b >> t )
1 3
b
h t (1 + 3 )
6
h
r
 
t
>> t )
1 h (1 + 3b / h )
2 t (1 + b / h ) 2
( h , b >> t
b
t
b (h o − h i )
hi
ho
≈ 2bt
( h , b >> t )
b
b 3
( h o − h 3i )
12
1
≈ b t h o2
2
t
h
2t
2 t (h + b)
( h , b >> t )
1 3
b
h t (1 + 3 )
6
h
3 h o2
2 bt
( h , b >> t )
1 h (1 + 3b / h )
2 t (1 + b / h ) 2
( h , b >> t )
b
Mike Ashby, 2013
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What values of ϕe exist in reality?
Data for structural steel, 6061 aluminium, pultruded GFRP and timber
ϕ e = 100
0 .0 1
ϕe =
12 I
A2
⇒ log ( I ) = 2 log ( A ) + log
1 e -0 0 3
ϕ e =1
ϕe
12
S te e l U n iv e rsal B e am
P u ltr u d e d G F R P I-se ctio n
P u ltr u d e d G F R P C h an n e l
4
Second
moment
area,I_max
I (m(m^4)
)
Second
Moment
of Areaof
(major),
1 e -0 0 4
P u ltr u d e d G F R P An g le
1 e -0 0 5
S te e l tu b e
G lu lam r e ctan g u lar
E xtr u d e d Al-C h an n e l
S te e l tu b e
1 e -0 0 6
E xtru d e d Al I-se ctio n
S o ftw o o d re ctan g u lar
1 e -0 0 7
P u ltru d e d G F R P tu b e
1 e -0 0 8
E xtru d e d Al-tu b e
1 e -0 0 9
E xtr u d e d Al-an g le
1 e -0 1 0
E xtru d e d Al A-an g le
1 e -0 1 1
1 e -0 0 5
1 e -0 0 4
1 e -0 0 3
0 .0 1
0 .1
S e c tio n Area,
A re a , A
Section
A (m
(m^22))
Mike Ashby, 2013
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Limits for Shape Factors ϕe and ϕf
There is an upper limit to shape factor for each material
Material
Max ϕe
Max ϕf
Steels
65
13
Aluminium alloys
44
10
GFRP and CFRP
39
9
Unreinforced polymers
12
5
Woods
8
3
Elastomers
<6
-
Other materials
..can calculate
Limit set by:
(a) manufacturing constraints
(b) local buckling
Modulus
Theoretical limit:
Mike Ashby, 2013
ϕe ≈ 2
E
σy
Yield strength
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Indices that include shape
Function
Constraint
Area A
Beam (shaped section).
Bending stiffness = S:
S=
CEI
3
L
I is the second moment of area:
ϕ e = 12
Objective
F
I
A2
1/ 2
 12 I 

A = 
 ϕe 
Minimise mass, m, where:
m = ALρ
1/ 2
 12 S L5 

m =


C


Mike Ashby, 2013


ρ


 (ϕ E )1 / 2 
 e

L
m = mass
A = area
L = length
ρ = density
b = edge length
S = stiffness
I = second moment of area
E = Youngs Modulus
Chose materials with smallest


ρ


 (ϕ E )1 / 2 
 e

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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 with low
Material
ρ, Mg/m3
E, GPa
ϕe,max
ρ
E1 / 2
ρ
(ϕ eE )1 / 2
(
)
ρ / E1 / 2
ρ / ϕ e,max E 1 / 2
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
8
0.25
0.088
Commentary:
13
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
Mike Ashby, 2013
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Shape on selection charts
Note that
ρ
1/ 2
(ϕ e E )
=
ρ / ϕe
1/ 2
(E / ϕ e )
ρ* = ρ / ϕe
New material with
E* = E / ϕe
Al: ϕe = 1
Al: ϕe = 44
ρ
E
1/ 2
=C
Mike Ashby, 2013
www.grantadesign.com/education/resources
So what?
When materials carry bending, torsion or axial compression, the section
shape becomes important.
The “shape efficiency” is 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 ρ / E1 / 2 ) guide the choice.
If materials can be made -- or are available -- in different shapes, then
indices which include the shape (eg ρ / (ϕ E )1 / 2 ) guide the choice.
Mike Ashby, 2013
www.grantadesign.com/education/resources
End of Unit 9
Mike Ashby, 2013
www.grantadesign.com/education/resources
Lecture Unit Series
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Mike Ashby, 2013
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Author
Mike Ashby
University of Cambridge, Granta Design Ltd.
www.grantadesign.com
www.eng.cam.ac.uk
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© M. F. Ashby, 2013
Granta’s Teaching Resources website aims to support teaching of materials-related courses in Engineering, Science and Design.
The resources come in various formats and are aimed at different levels of student.
This resource is part of a set created by Mike Ashby to help introduce materials and materials selection to students.
The website also contains other resources contributed by faculty at the 800+ universities and colleges worldwide using Granta’s CES EduPack.
The teaching resource website contains both resources that require the use of CES EduPack and those that don’t.
Mike Ashby, 2013
www.grantadesign.com/education/resources
www.grantadesign.com/education/resources