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

They depend on shapes

Moment of area about axis x dA 2 Ixx = y dA ∫A

Moment of area about axis y 2 Iyy = x dA ∫A

Polar moment of area J = r 2 dA ∫A Modes of loading: Axial loading

F σ = Stress A δ ε = Strain L

• From Hooke’s Law (linearly elastic material): σ F σ = Eε ⇒ ε = = E AE

F • From the definition of strain: δ FL ε = ⇒ δ = L AE

σσσ F AE S = = Stiffness δ L

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:

L′ = (ρ − y)θ δ = L' − L = ()ρ − y θ − ρθ = −yθ δ yθ y ε = = − = − Strain (varies linearly) z L ρθ ρ z c ⇒ c εmax = ρ = ρ εmax y ε = − ε z c max x Modes of loading: Bending

For a linearly elastic material: y σ = Eε = − Eε z z c max y = − σ Stress (varies linearly) c max

M M (c = y ) σmax = = max Z Ixx /c I σz Z = xx Bending strength modulus c

M I = moment of area σ z = XX Ixx /y about the bending axis

Modes of loading: Bending

 d2y  F F CEIxx E I = M()z  S = = Stiffness  xx 2  3  dz  z δ L δ I XX = moment of area about the bending axis

C = constant (depending on the loading conditions) Modes of loading: Torsion

Torsion : Prismatic members subjected T to twisting couples or torques

Consider an interior section of the shaft. As a torsional load is applied, the shear strain is equal to angle of twist. ) ) ρφ A 'A = Lγ = ρφ ⇒ γ = L cφ ρ γ = γ = γ max L c max Shear strain ( ∝ twist angle and radius)

Modes of loading: Torsion

For a linearly elastic material: T ρ τ = Gγ = Gγ c max ρ = τ Shear stress (varies linearly) c max

T T τ = = (c = ρmax ) max Q K/c K Q = Twisting strength modulus c

T τ = K = torsional moment of area K/ ρ Modes of loading: Torsion

T KG  Lγ L τ L Tρ  Stiffness φ = = =  ST = =  ρ ρ G ρG K  T φ L

• 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

T T τ = = (c = ρmax ) max Q J/c T τ = J = polar moment of area J/ ρ

T JG S = = Stiffness T φ L

Modes of loading: Buckling

F F 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 F σ = ≤ σ A y - deformation falls within specifications

FL δ = ≤ δ AE lim

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

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

k(2∆θ ) = restoring moment L L F sin ∆θ = F ∆θ = destabiliz ing moment 2 2

• Column is stable (tends to return to F’ aligned orientation) if L F ∆θ < k()2∆θ 2 4k F’ F < F = cr L

Modes of loading: Buckling

F The critical loading is calculated from Euler’s formula

2 π EI min Fcr = L2

Stress corresponding to critical loading L 2 π E ( r 2 = I /A inertia radius ) σcr = min ()L r 2 2 π E ( λ2 = L2/r 2 slendernes s ) σcr = λ2 Modes of loading: Buckling

2 π EI min Fcr = F F 2 F F Le

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

Shape efficiency

“Shape ” = cross section formed to a tubes I-sections hollow box-section sandwich 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

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

Area A matters, not shape

Area A and shape I ( XX ) matter

Area A and shape (J, K) matter

Area A and shape I ( min ) matter

Shape and mode of loading

Function Tie-rod F F Area A L

Objective Minimise mass m: m = A L ρρρ m = mass A = area L = length Constraints Stiffness of the tie S: ρ = density S = stiffness E A E = Youngs Modulus S = L

Area A matters, not shape Shape and mode of loading

Function Tie-rod F F Area A L

Objective Minimise mass m: m = A L ρρρ m = mass A = area L = length Constraints Must not fail under load F: ρ = density σy= yield strength F/A < σσσy

Area A matters, not shape

Shape and mode of loading F Function Beam (solid square section). b

b Objective Minimise mass , m, where: m = ALρ = b2 Lρ L m = mass A = area Constraint Stiffness of the beam S: L = length ρ = density CEI S = b = edge length 3 S = stiffness L I = second moment of area E = Youngs Modulus Area A and shape matter

b4 I is the second moment of area: I= 12 Shape and mode of loading F Function Beam (solid square section). b

b Objective Minimise mass , m, where: m = ALρ = b2 Lρ L

m = mass Constraint Must not fail under load F A = area L = length M M⋅b/2 ρ = density σ y > = b = edge length Z I I = second moment of area

σy = yield strength Area A and shape matter

b4 I is the second moment of area: I= 12

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

Take ratio of bending stiffness S of shaped section to that (S o) of a neutral reference section of the same cross-section area • Define a standard reference section: a solid square with area A = b 2 (alternatively: solid circular section) • Second moment of area is I; stiffness scales as E I (S = CE I/L 3) F b C IE b S = L3 L Shape efficiency: Bending stiffness

Area A is b4 A2 I = = constant o 12 12 Area A = b 2 Area A and b modulus E b unchanged

Define shape factor for elastic bending, measuring efficiency, as S IE I φ e = = = 12 B 2 So EIo A

Shape efficiency: Bending stiffness Properties of Shape Factor

The shape factor is dimensionless -- a pure number

It characterizes shape

Increasing size at constant shape

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

Tabulation of Shape Factors

S EI I φ e = = = 12 B 2 So EIo A

(standard reference section: solid square section)

b4 A2 I = = o 12 12 Tabulation of Shape Factors

S IE I φ e = = = 4π B 2 So EIo A

(standard reference section: solid circular section) π r 4 A 2 I = = o 4 4π

Shape efficiency: Bending strength

• 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 (M f) scales as σ y Z (M f = σy Z) I Z = y Area A is max constant b4 2 b3 A3/2 Z = ⋅ = = o 12 b 6 6 Area A and yield Area A = b 2 strength σ y unchanged b b

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

M σ Z Z φ f = f = y = 6 B 3/2 Mfo σ yZo A Shape efficiency: Bending strength

Shape efficiency: Twisting stiffness

Take ratio of twisting stiffness S T of shaped section to that (ST,o ) of a reference section ( solid square ) of the same cross-section area T KG S = = T θ L • Torsional moment of area is K (= J for circular sections) ; stiffness scales as KG

b⋅h3  b  K = ⋅1− 0,58  = 0,14 A 2 o 3  h  b = h Area A = b 2 Area A and b modulus G b unchanged

Define shape factor for elastic twisting, measuring efficiency, as S KG K φ e = T = = 7,14 T 2 ST, o KoG A Shape efficiency: Twisting stiffness

Shape efficiency: Twisting strength

• 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 τ Q (Tf = τ Q) J Q = (for circular rmax sections only)

b2h2 b3 A3/2 Q = = = o 3h +1,8b 4,8 4,8 Area A and b = h strength τ Area A = b 2 unchanged b b

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

T τ Q Q φ f = f = = 4,8 T 3/2 Tf, o τ Qo A 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

2 2 • Critical load (Fcr ) scales as EI min (Fcr = π EI min /L e ) e • The shape factor is the same as that for elastic bending (φ B ), with I replaced by Imin

b4 A 2 I = I = = min, o o 12 12 Area A = b 2 Area A and b modulus E b unchanged

Define shape factor for resistance to buckling, measuring efficiency, as F IE I φ = cr = min = 12 min Bck 2 Fcr, o EImin, o A 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

Two types of limit for shape factors - manufacturing constraints (processability of materials) - mechanical stability of shaped sections 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.

Local buckling Modulus

  e E φ f ≈ φ e  Theoretical limit: φ B ≈ 2.3  B B  σy Yield strength

e What values of φφφB exist in reality?

I φ e = 12 ⇒ B A 2  e  φ B log ()()I = 2log A + log    12 

Slope = 2

C IE xx x S = L3

Ixx > Ixx > Ixx x f What values of φφφB exist in reality?

Z φ f = 6 ⇒ B A 3/2  f  3 φ B log ()Z = log ()A +log   2  6 

Slope = 3/2

Ixx x Z = ymax

Ixx > Ixx > Ixx x

e What values of φφφB exist in reality?

Data for structural steel, 6061 aluminium, pultruded GFRP and wood e ϕ =100 φB = ϕe e

0.01 ϕ =1 12 I ϕe e ϕe = ⇒ log (I)=2log (A)+log A2 12 Steel Universal Beam 1e-003 Pultruded GFRP I-section )

4 Pultruded GFRP Channel 1e-004 Slope = 2 Pultruded GFRP Angle

1e-005 Steel tube Glulam rectangular

Extruded Al-Channel Steel tube 1e-006 Extruded Al I-section Softwood rectangular 1e-007

Pultruded GFRP tube

1e-008 Extruded Al-tube

Second moment Second (m I area, of 1e-009 Second Moment of Area (major), I_max (m^4) I_max (major), Area of SecondMoment Extruded Al-angle 1e-010

Extruded Al A-angle 1e-011 1e-005 1e-004 1e-003 0.01 0.1 SectionSection Area, Area, AA (m^2) (m 2) Indices that include shape

F Function Beam (shaped section) Area A

Objective Minimise mass , m, where: m = ALρ L Constraint Bending stiffness of the beam S: 3 CEI SL S = I = m = mass L3 CE A = area I is the second moment of area: L = length /1 2 ρ = density I 12 I  b = edge length ϕ = 12 A =   e 2   S = stiffness A  ϕe  I = second moment of area E = Youngs Modulus Combining the equations gives:

5 /1 2   12 S L   ρ   ρ  m =     Chose materials with smallest    /1 2   /1 2   C   ()ϕeE   ()ϕeE 

Selecting material-shape combinations

Materials for stiff, shaped beams of minimum weight ρ • Fixed shape ( ϕ fixed): choose materials with low e E /1 2 ρ • Shape ϕe a variable: choose materials and shapes with low /1 2 ()ϕeE

Material ρ, Mg/m 3 E, GPa ϕ 2/1 /1 2 e,max ρ/E ρ /(ϕ ,e max E) 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

• 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 Selecting material-shape combinations

I φ e = 12 ⇒ B A 2  e  φ B log ()()I = 2log A + log    12 

Selecting material-shape combinations

S L = E I ⇒

log (E ) = - log (I)+ log (S L ) Selecting material-shape combinations

m/L = ρ A ⇒ log (A ) = - log (ρ)+ log (m/L )

Selecting material-shape combinations

Required section stiffness: EI = 10 6 N.m 2

Shape factor: e φB = 10 Selecting material-shape combinations

Required section stiffness: EI = 10 6 N.m 2

Shape factor: e φB = 10

Selecting material-shape combinations

Required section stiffness: EI = 10 6 N.m 2

Shape factor: e φB = 2 Selecting material-shape combinations

Required section stiffness: EI = 10 6 N.m 2

Shape factor: e φB = 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

Selection with variable shape

Shape S

Selecting material-shape combinations

four

4 S

• When the groups are separable, the optimum choice of material and shape becomes independent of the detail of the design. It is the same for all geometries G and all values of functional requirements F. • The performance for all F and G is maximized by maximizing

f3(M) and f 4(S). Selecting material-shape combinations

four

4 S

• In theory f 4(S) is independent of the material (shape factors depend on shape only). • In reality the shape factors depend on material (because of constraints from material-process-shape relations, and limits from processability and mechanical behaviour of material which form the shape), therefore . f3(M) f 4(S) constitutes the new performance index. • Shaped material can be considered as a new material with modified (improved) properties.

Shape on selection charts

ρ ρ/ϕ ρ* ρ* = ρ/ ϕe Note that = e = New material with E 1/2 E/ 1/2 E* 1/2 ()ϕe ()ϕe () E* = E / ϕe

Silicon Carbide Tungsten Carbides 1000 Boron Carbide Alumina Al: ϕe = 1 Silicon Steels Nickel alloys Al alloys Copper alloys 100 Mg alloys Zinc alloys CFRP Bamboo GFRP Titanium Wood 10 Concrete Lead alloys Plywood PET PVC PP PUR 1 PE Rigid Polymer Foams PTFE

0.1 EVA

Silicone Young's Modulus (typical) (GPa) (typical) ModulusYoung's Cork Young’s Young’s modulus(GPa) 0.0 1 Flexible Polymer Foams Polyisoprene ρ Polyurethane = C 1e-003 Butyl Rubber E /1 2 Neoprene

1e-004 0.01 0.1 1 10 Density (typical) (Mg/m^3) Density (Mg/m 3) Shape on selection charts

ρ ρ/ϕ ρ* ρ* = ρ/ ϕe Note that = e = New material with E 1/2 E/ 1/2 E* 1/2 ()ϕe ()ϕe () E* = E / ϕe

Silicon Carbide Tungsten Carbides 1000 Boron Carbide Alumina Al: ϕe = 1 Silicon Steels Nickel alloys Al alloys Copper alloys 100 Mg alloys Zinc alloys CFRP Bamboo Al: ϕ = 44 GFRP Titanium e Wood 10 Concrete Lead alloys Plywood PET E /44 Al PVC PP PUR 1 PE Rigid Polymer Foams PTFE

0.1 EVA

1e-004 0.01 0.1 1 10 ρ /44 Density (typical) (Mg/m^3) Al Density (Mg/m 3)

Data organisation: Structural sections

Standard prismatic sections

Kingdom Family Material and Member Attributes

Material properties • Angles Extruded Al alloy ρ, E, σy • Channels Pultruded GFRP Dimensions A ... Structural • I-sections Section properties: Structural steel sections • Rectangular I, Z, K, Q ... Softwood • T-sections Structural properties: • Tubes EI, σ y Z, GK ...

A record Part of a record for a structural section

Pultruded GFRP Vinyl Ester (44 x 3.18)

Material properties Price 3.99 - 4.87 GBP/kg Density 1.65 - 1.75 Mg/m^3 Young's Modulus17 - 18 GPa Yield Strength 195 - 210 MPa Dimensions Diameter, B 0.0439 - 0.0450 m Thickness, t 2.54e-3 - 3.81e-3 m Section properties Section Area, A 3.3e-004 - 4.93e-004 m^2 Second 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 properties Mass 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.m Etc.

Example: Selection of a beam

Specification B x D F

Function Beam

Required stiffness: Constraint 5 2 L EI max > 10 N.m m = mass/unit length Required strength: a Ca = cost/unit length 3 σyZ > 10 N.m D = beam depth B = width Dimension I = second moment of area E = Young’s modulus B < 100 mm Z = section modulus D < 200 mm σy = yield strength

Objectives (a) Find lightest beam (b) Find cheapest beam Applying constraints with a limit stage

Dimensions Minimum Maximum

Depth D 200 mm

Width B 100 mm

Section attributes

Bending Stiffness E.I100,000 N.m 2

Failure Moment Y. Z 1000 N.m

Optimisation: Minimising mass/length

1e+009

Steel Rect.Hollow 1e+008 Selection box Steel Universal Beam

) 1e+007 2

5 2 Steel Equal Angle E.I max = 10 Nm (Nm 1e+006

max Extruded Al-tube

100000 Steel tube Extruded Al I-section

10000

Extruded Al Angle

1000 Bending Bending StiffnessE.I

100 Bending Stiffness (major), E.I_max (N.m^2) E.I_max (major), Bending Stiffness

Pultruded GFRP tube 10 Bending Stiffness EI vs. mass per unit length

1 0.1 1 10 100 1000 Mass per per unit unit length, length m/l (kg/m) Results: Selection of a beam

OUTPUT: objective – minimum weight

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)

OUTPUT: objective – minimum cost 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 (egρ /E /1 2 ) guide the choice.

If materials can be made -- or are available -- in different shapes, 2/1 then indices which include the shape (egρ /(φE) ) guide the choice.

The CES Structural Sections database allows standard sections to be explored and selected.