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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
Modes of loading: Bending
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=
Modes of loading: Bending
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
))
⇒
∝
Modes of loading: Torsion
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 =
Modes of loading: Torsion
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
Modes of loading: Buckling
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
Modes of loading: Buckling
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
Modes of loading: Buckling
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
Shape and mode of loading
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
Shape and mode of loading
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
Shape and mode of loading
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
Shape and mode of loading
m = mass
A = area
L = length
ρ = densityb = edge length
I = second moment of area
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
Minimise mass, m, where:
Function
Objective
Constraint
I is the second moment of area:12
bI
4
=
Area A and shape matter
Shape and mode of loading
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 =
Shape efficiency: Bending stiffness
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
Shape efficiency: Bending stiffness
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 ==
Tabulation of Shape Factors
(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
Tabulation of Shape Factors
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
I = second moment of area
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
Minimise mass, m, where:
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
Selecting material-shape combinations
⇒=φ2
eB
A
I12
( ) ( )
φ+=
12
logA2logIlog
eB
Selecting material-shape combinations
⇒= I ES L
( ) ( ) ( )LS logIlog-Elog +=
Selecting material-shape combinations
⇒ρ= Am/L
( ) ( ) ( )m/L loglog-Alog +ρ=
Selecting material-shape combinations
Required section stiffness:
EI = 106 N.m2
Shape factor:
φBe = 10
Selecting material-shape combinations
Required section stiffness:
EI = 106 N.m2
Shape factor:
φBe = 10
Selecting material-shape combinations
Required section stiffness:
EI = 106 N.m2
Shape factor:
φBe = 2
Selecting material-shape combinations
Required section stiffness:
EI = 106 N.m2
Shape factor:
φBe = 30
Selecting material-shape combinations
Required section strength:
σyZ > Vmin
Selecting material-shape combinations
Required section strength:
σyZ > Vmin
Selecting material-shape combinations
Selection with fixed shape
Selecting material-shape combinations
Shape
S
Selection with variable shape
Selecting material-shape combinations
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).
Selecting material-shape combinations
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
I = second moment of area
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)
Extruded aluminium box section, YS 255 MPa (135 x 35 x 4.0 mm)
Extruded aluminium box section, YS 255 MPa (152 x 44 x 3.2 mm)
Extruded aluminium box section, YS 255 MPa (152 x 64 x 3.2 mm)
Sawn softwood, rectangular section (150 x 36)
Sawn softwood, rectangular section (150 x 38)
Sawn softwood, rectangular section (175 x 32)
Sawn softwood, rectangular section (200 x 22)
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